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Mathematics LibreTexts

8-8. Vectors

Vectors
In this section you will:
  • View vectors geometrically.
  • Find magnitude and direction.
  • Perform vector addition and scalar multiplication.
  • Find the component form of a vector.
  • Find the unit vector in the direction of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi></mrow></annotation-xml></semantics></math>.
  • Perform operations with vectors in terms of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>i</mi></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>j</mi></mrow></annotation-xml></semantics></math>.
  • Find the dot product of two vectors.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in [link]. What are the ground speed and actual bearing of the plane?

<figure class="small" id="Figure_08_08_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;">Image of a plan flying SE at 140 degrees and the north wind blowing </figure>

Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

A Geometric View of Vectors

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:

  • Lower case, boldfaced type, with or without an arrow on top such as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>u</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>w</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mover accent="true"/></mrow></annotation-xml></semantics></math> v → ,   u → ,  w → .
  • Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>a vector can be represented as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mover accent="true"/></mrow></annotation-xml></semantics></math> PQ →  .  The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
  • Given an initial point of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a,b ), a vector may be represented as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.

This last symbol <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩ has special significance. It is called the standard position. The position vector has an initial point <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and a terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mover accent="true"/></mrow></annotation-xml></semantics></math> CD →  is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 1 , y 1 ) and the terminal point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 , y 2 ), then the position vector is found by calculating

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow></mover></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> →  = ⟨ x 2 − x 1 , y 2 − y 1 ⟩           = ⟨ a,b ⟩

In [link], we see the original vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mover accent="true"/></mrow></annotation-xml></semantics></math> CD →  and the position vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mover accent="true"/></mrow></annotation-xml></semantics></math> AB → .

<figure class="small" id="Figure_08_08_003">Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.</figure>
Properties of Vectors

A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and is identified by its terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.

Find the Position Vector

Consider the vector whose initial point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,3 ) and terminal point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6,4 ). Find the position vector.

The position vector is found by subtracting one x-coordinate from the other x-coordinate, and one y-coordinate from the other y-coordinate. Thus

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 6−2,4−3 ⟩    =⟨ 4,1 ⟩

The position vector begins at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and terminates at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,1 ). The graphs of both vectors are shown in [link].

<figure class="small" id="Figure_08_08_022">Plot of the original vector in blue and the position vector in orange extending from the origin.</figure>

We see that the position vector is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 4,1 ⟩.

Drawing a Vector with the Given Criteria and Its Equivalent Position Vector

Find the position vector given that vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has an initial point at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −3,2 ) and a terminal point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,5 ), then graph both vectors in the same plane.

The position vector is found using the following calculation:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 4−(−3),5−2 ⟩   =⟨ 7,3 ⟩

Thus, the position vector begins at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and terminates at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,3 ). See [link].

<figure class="small" id="Figure_08_08_004">Plot of the two given vectors their same position vector.</figure>

Draw a vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>that connects from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.

Finding Magnitude and Direction

To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

Magnitude and Direction of a Vector

Given a position vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi></mrow></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩,the magnitude is found by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |= a 2 + b 2 .The direction is equal to the angle formed with thex-axis, or with the y-axis, depending on the application. For a position vector, the direction is found by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> b a )⇒θ= tan −1 ( b a ), as illustrated in [link].

<figure class="small" id="Figure_08_08_017">Standard plot of a position vector (a,b) with magnitude |v| extending into Q1 at theta degrees. </figure>

Two vectors v and u are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.

Finding the Magnitude and Direction of a Vector

Find the magnitude and direction of the vector with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −8,1 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,−5 ).Draw the vector.

First, find the position vector.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −2,−(−8),−5−1 ⟩   =⟨ 6,−6 ⟩

We use the Pythagorean Theorem to find the magnitude.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><mi>u</mi><mo>|</mo><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (6) 2 + (−6) 2       = 72       =6 2

The direction is given as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −6 6 =−1⇒θ= tan −1 (−1)             =−45°

However, the angle terminates in the fourth quadrant, so we add 360° to obtain a positive angle. Thus,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>45°</mn><mo>+</mo><mn>360°</mn><mo>=</mo><mn>315°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>See[link].

<figure class="small" id="Figure_08_08_018">Plot of the position vector extending into Q4 from the origin with the magnitude 6rad2.</figure>
Showing That Two Vectors Are Equal

Show that vector v with initial point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,−3 ) and terminal point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,2 ) is equal to vector u with initial point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,−3 ) and terminal point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −7,2 ). Draw the position vector on the same grid as v and u. Next, find the magnitude and direction of each vector.

As shown in [link], draw the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>starting at initial<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,−3 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,2 ). Draw the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,−3 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −7,2 ). Find the standard position for each.

Next, find and sketch the position vector for v and u. We have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mo>⟨</mo><mn>−1</mn><mo>−</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>   =⟨−6,5⟩ u=⟨−7−(−1),2−(−3)⟩   =⟨−6,5⟩

Since the position vectors are the same, v and u are the same.

An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><mi>v</mi><mo>|</mo><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (−1−5) 2 + (2−(−3)) 2       = (−6) 2 + (5) 2       = 36+25       = 61 |u|= (−7−(−1)) 2 + (2−(−3)) 2       = (−6) 2 + (5) 2       =36+25       = 61

As the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector gives

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5 6 ⇒θ= tan −1 ( − 5 6 )             =−39.8°

However, we can see that the position vector terminates in the second quadrant, so we add<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>180°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Thus, the direction is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>39.8°</mn><mo>+</mo><mn>180°</mn><mo>=</mo><mn>140.2°</mn><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_08_08_005">Plot of the two given vectors their same position vector.</figure>

Performing Vector Addition and Scalar Multiplication

Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>u</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> x,y ⟩as an arrow or directed line segment from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>vectors can be situated anywhere in the plane. The sum of two vectors u and v, or vector addition, produces a third vector u+ v, the resultant vector.

To find u + v, we first draw the vector u, and from the terminal end of u, we drawn the vector v. In other words, we have the initial point of v meet the terminal end of u. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in [link].

<figure class="small" id="Figure_08_08_008">Diagrams of vector addition and subtraction. </figure>

Vector subtraction is similar to vector addition. To find uv, view it as u + (−v). Adding −v is reversing direction of v and adding it to the end of u. The new vector begins at the start of u and stops at the end point of −v. See [link] for a visual that compares vector addition and vector subtraction using parallelograms.

<figure class="small" id="Figure_08_08_009">Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.</figure>
Adding and Subtracting Vectors

Given <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>u</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 3,−2 ⟩and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −1,4 ⟩,find two new vectors u + v, and uv.

To find the sum of two vectors, we add the components. Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>+</mo><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3,−2 ⟩+⟨ −1,4 ⟩             =⟨ 3+(−1),−2+4 ⟩             =⟨ 2,2 ⟩

See [link](a).

To find the difference of two vectors, add the negative components of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Thus,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3,−2 ⟩+⟨ 1,−4 ⟩                       =⟨ 3+1,−2+(−4) ⟩                       =⟨ 4,−6 ⟩

See [link](b).

<figure class="medium" id="Figure_08_08_019"> <figcaption>(a) Sum of two vectors (b) Difference of two vectors</figcaption> Further diagrams of vector addition and subtraction.</figure>

Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

Scalar Multiplication

Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩ by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi></mrow></annotation-xml></semantics></math>, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> ka,kb ⟩

Only the magnitude changes, unless<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is negative, and then the vector reverses direction.

Performing Scalar Multiplication

Given vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 3,1 ⟩, find 3v, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and −v.

See [link] for a geometric interpretation. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 3,1 ⟩,then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mn>3</mn><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3⋅3,3⋅1 ⟩         =⟨ 9,3 ⟩   1 2 v=⟨ 1 2 ⋅3, 1 2 ⋅1 ⟩         =⟨ 3 2 , 1 2 ⟩ −v=⟨ −3,−1 ⟩
<figure class="small" id="Figure_08_08_007">Showing the effect of scaling a vector: 3x, 1x, .5x, and -1x. The 3x is three times as long, the 1x stays the same, the .5x halves the length, and the -1x reverses the direction of the vector but keeps the length the same. The rest keep the same direction; only the magnitude changes.</figure>
Analysis

Notice that the vector 3v is three times the length of v, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mfrac><mn>1</mn></mfrac></annotation-xml></semantics></math> 2<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi><mtext> </mtext></annotation-xml></semantics></math>is half the length of v, and –v is the same length of v, but in the opposite direction.

Find the scalar multiple 3<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>u</mi></annotation-xml></semantics></math> given <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>u</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 5,4 ⟩.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 15,12 ⟩

Using Vector Addition and Scalar Multiplication to Find a New Vector

Given <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>u</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 3,−2 ⟩and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −1,4 ⟩,find a new vector w = 3u + 2v.

First, we must multiply each vector by the scalar.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>3</mn><mi>u</mi><mo>=</mo><mn>3</mn><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3,−2 ⟩       =⟨ 9,−6 ⟩ 2v=2⟨ −1,4 ⟩       =⟨ −2,8 ⟩

Then, add the two together.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>w</mi><mo>=</mo><mn>3</mn><mi>u</mi><mo>+</mo><mn>2</mn><mi>v</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>      =⟨ 9,−6 ⟩+⟨ −2,8 ⟩      =⟨ 9−2,−6+8 ⟩      =⟨ 7,2 ⟩

So, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>w</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 7,2 ⟩.

Finding Component Form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>direction, and the vertical component is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>direction. For example, we can see in the graph in [link] that the position vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 2,3 ⟩comes from adding the vectors v1 and v2. We have v1with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,0 ). 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 =⟨ 2−0,0−0 ⟩       =⟨ 2,0 ⟩

We also have v2 with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 3 ). 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 =⟨ 0−0,3−0 ⟩        =⟨ 0,3 ⟩

Therefore, the position vector is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2+0,3+0 ⟩    =⟨ 2,3 ⟩

Using the Pythagorean Theorem, the magnitude of v1 is 2, and the magnitude of v2 is 3. To find the magnitude of v, use the formula with the position vector.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><mi>v</mi><mo>|</mo><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> | v 1 | 2 +| v 2 | 2      = 2 2 + 3 2      = 13

The magnitude of v is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msqrt/></mrow></annotation-xml></semantics></math> 13 . To find the direction, we use the tangent function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> y x .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> v 2 v 1 tan θ= 3 2          θ= tan −1 ( 3 2 )=56.3°
<figure class="small" id="Figure_08_08_020">Diagram of a vector in root position with its horizontal and vertical components.</figure>

Thus, the magnitude of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msqrt/></mrow></annotation-xml></semantics></math> 13  and the direction is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> 56.3 ∘off the horizontal.

Finding the Components of the Vector

Find the components of the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,2 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,4 ).

First find the standard position.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 7−3,4−2 ⟩    =⟨ 4,2 ⟩

See the illustration in [link].

<figure class="small" id="Figure_08_08_021">Diagram of a vector in root position with its horizontal (4,0) and vertical (0,2) components.</figure>

The horizontal component is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><msub><mi>v</mi></msub></annotation-xml></semantics></math> 1<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 4,0 ⟩ and the vertical component is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><msub/></annotation-xml></semantics></math> v 2<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 0,2 ⟩.

Finding the Unit Vector in the Direction of v

In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.

Unit vectors are defined in terms of components. The horizontal unit vector is written as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>i</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mo>=</mo><mrow><mo>⟨</mo></mrow></annotation-xml></semantics></math> 1,0 ⟩and is directed along the positive horizontal axis. The vertical unit vector is written as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>j</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mo>=</mo><mrow><mo>⟨</mo></mrow></annotation-xml></semantics></math> 0,1 ⟩and is directed along the positive vertical axis. See[link].

<figure class="small" id="Figure_08_08_011">Plot showing the unit vectors i=91,0) and j=(0,1)</figure>
The Unit Vectors

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a nonzero vector, then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> v | v |  is a unit vector in the direction of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.

Finding the Unit Vector in the Direction of v

Find a unit vector in the same direction as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mo>=</mo><mrow><mo>⟨</mo></mrow></annotation-xml></semantics></math> −5,12 ⟩.

First, we will find the magnitude.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><mi>v</mi><mo>|</mo><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (−5) 2 + (12) 2       = 25+144       = 169       =13

Then we divide each component by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |, which gives a unit vector in the same direction as v:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>v</mi></mfrac></mrow></annotation-xml></semantics></math> | v | =− 5 13 i+ 12 13 j

or, in component form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>v</mi></mfrac></mrow></annotation-xml></semantics></math> | v | =⟨ − 5 13 , 12 13 ⟩

See [link].

<figure class="small" id="Figure_08_08_012">Plot showing the unit vector (-5/13, 12/13) in the direction of (-5, 12)</figure>

Verify that the magnitude of the unit vector equals 1. The magnitude of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 13 i+ 12 13 j is given as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></msqrt></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> − 5 13 ) 2 + ( 12 13 ) 2 = 25 169 + 144 169                             = 169 169 =1

The vector u<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 13i<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 13j is the unit vector in the same direction as v<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −5,12 ⟩.

Performing Operations with Vectors in Terms of i and j

So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of i and j.

Vectors in the Rectangular Plane

Given a vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 1 , y 1 ) and terminal point <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub/></mrow></annotation-xml></semantics></math> x 2 , y 2 ), v is written as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 − x 1 )i+( y 1 − y 2 )j

The position vector from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a,b ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 − x 1 )=a and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> y 2 − y 1 )=b, is written as v = ai + bj. This vector sum is called a linear combination of the vectors i and j.

The magnitude of v = ai + bj is given as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |= a 2 + b 2 . See [link].

<figure class="small" id="Figure_08_08_010">Plot showing vectors in rectangular coordinates in terms of i and j. The position vector v (in orange) extends from the origin to some point (a,b) in Q1. The horizontal (ai) and vertical (bj) components are shown.</figure>
Writing a Vector in Terms of i and j

Given a vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,−6 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −6,6 ), write the vector in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 − x 1 )i+( y 2 − y 1 )j    =(−6−2)i+(6−(−6))j    =−8i+12j
Writing a Vector in Terms of i and j Using Initial and Terminal Points

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( −1,3 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( 2,7 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 − x 1 )i+( y 2 − y 1 )j v=(2−(−1))i+(7−3)j   =3i+4j

Write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,6 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Q</mi><mo>=</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,−5 ) in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo>=</mo><mn>8</mn><mi>i</mi><mo>−</mo><mn>11</mn><mi>j</mi></mrow></annotation-xml></semantics></math>

Performing Operations on Vectors in Terms of i and j

When vectors are written in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

Adding and Subtracting Vectors in Rectangular Coordinates

Given v = ai + bj and u = ci + dj, then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mi>v</mi><mo>+</mo><mi>u</mi><mo>=</mo><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> a+c )i+( b+d )j v−u=( a−c )i+( b−d )j
Finding the Sum of the Vectors

Find the sum of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> v 1 =2i−3j and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> v 2 =4i+5j.

According to the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 + v 2 =(2+4)i+(−3+5)j                  =6i+2j

Calculating the Component Form of a Vector: Direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.

Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v | replacing<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Vector Components in Terms of Magnitude and Direction

Given a position vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> x,y ⟩ and a direction angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>,</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x |v| and sin θ= y |v|         x=|v|cos θ        y=|v|sin θ

Thus,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mi>x</mi><mi>i</mi><mo>+</mo><mi>y</mi><mi>j</mi><mo>=</mo><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |cos θi+| v |sin θj, and magnitude is expressed as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |= x 2 + y 2 .

Writing a Vector in Terms of Magnitude and Direction

Write a vector with length 7 at an angle of 135° to the positive x-axis in terms of magnitude and direction.

Using the conversion formulas<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |cos θi and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |sin θj, we find that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mn>7</mn><mi>cos</mi><mo stretchy="false">(</mo><mn>135°</mn><mo stretchy="false">)</mo><mi>i</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>    =− 7 2 2 y=7sin(135°)j    = 7 2 2

This vector can be written as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mn>7</mn><mi>cos</mi><mo stretchy="false">(</mo><mn>135°</mn><mo stretchy="false">)</mo><mi>i</mi><mo>+</mo><mn>7</mn><mi>sin</mi><mo stretchy="false">(</mo><mn>135°</mn><mo stretchy="false">)</mo><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>or simplified as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 2 2 i+ 7 2 2 j

A vector travels from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,5 ). Write the vector in terms of magnitude and direction.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> 34 cos(59°)i+ 34 sin(59°)j

Magnitude =<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msqrt/></mrow></annotation-xml></semantics></math> 34

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> tan −1 ( 5 3 )=59.04°

Finding the Dot Product of Two Vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.

The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.

Dot Product

The dot product of two vectors<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> c,d ⟩ is the sum of the product of the horizontal components and the product of the vertical components.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>⋅</mo><mi>u</mi><mo>=</mo><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow></annotation-xml></semantics></math>

To find the angle between the two vectors, use the formula below.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> v | v | ⋅ u | u |
Finding the Dot Product of Two Vectors

Find the dot product of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 5,12 ⟩ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,4 ⟩.

Using the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo>⋅</mo><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5,12 ⟩⋅⟨ −3,4 ⟩           =5⋅(−3)+12⋅4           =−15+48           =33
Finding the Dot Product of Two Vectors and the Angle between Them

Find the dot product of v1 = 5i + 2j and v2 = 3i + 7j. Then, find the angle between the two vectors.

Finding the dot product, we multiply corresponding components.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 ⋅ v 2 =⟨ 5,2 ⟩⋅⟨ 3,7 ⟩                 =5⋅3+2⋅7                 =15+14                 =29

To find the angle between them, we use the formula<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> v |v| ⋅ u |u| .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> v |v| ⋅ u |u| =⟨ 5 29 + 2 29 ⟩⋅⟨ 3 58 + 7 58 ⟩                                          = 5 29 ⋅ 3 58 + 2 29 ⋅ 7 58                                         = 15 1682 + 14 1682 = 29 1682                                          =0.707107 cos −1 (0.707107)=45°

See [link].

<figure class="small" id="Figure_08_08_014">Plot showing the two position vectors (3,7) and (5,2) and the 45 degree angle between them.</figure>
Finding the Angle between Two Vectors

Find the angle between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,4 ⟩ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 5,12 ⟩.

Using the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>θ</mi><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos −1 ( u |u| ⋅ v |v| ) ( u |u| ⋅ v |v| )= −3i+4j 5 ⋅ 5i+12j 13                       =( − 3 5 ⋅ 5 13 )+( 4 5 ⋅ 12 13 )                      =− 15 65 + 48 65                       = 33 65                    θ= cos −1 ( 33 65 )                       = 59.5 ∘

See [link].

<figure class="small" id="Figure_08_08_013">Plot showing the two position vectors (-3,4) and (5,12) and the 59.5 degree angle between them.</figure>
Finding Ground Speed and Bearing Using Vectors

We now have the tools to solve the problem we introduced in the opening of the section.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See [link].

<figure class="small" id="Figure_08_08_015">Image of a plan flying SE at 140 degrees and the north wind blowing.</figure>

The ground speed is represented by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the diagram, and we need to find the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in order to calculate the adjusted bearing, which will be<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mn>140°</mn><mo>+</mo><mi>α</mi><mtext> </mtext><mo>.</mo></mrow></annotation-xml></semantics></math>

Notice in [link], that angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>C</mi><mi>O</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>must be equal to angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>O</mi><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the rule of alternating interior angles, so angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>C</mi><mi>O</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is 140°. We can find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the Law of Cosines:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>x</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 = (16.2) 2 + (200) 2 −2(16.2)(200)cos(140°) x 2 =45,226.41    x= 45,226.41    x=212.7

The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin α 16.2 = sin(140°) 212.7                           sin α= 16.2sin(140°) 212.7                                      =0.04896 sin −1(0.04896)=2.8°

Therefore, the plane has a SE bearing of 140°+2.8°=142.8°. The ground speed is 212.7 miles per hour.

Access these online resources for additional instruction and practice with vectors.

Key Concepts

  • The position vector has its initial point at the origin. See [link].
  • If the position vector is the same for two vectors, they are equal. See [link].
  • Vectors are defined by their magnitude and direction. See [link].
  • If two vectors have the same magnitude and direction, they are equal. See [link].
  • Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See[link].
  • Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See [link] and [link].
  • Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See [link].
  • The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
  • The magnitude of a vector in the rectangular coordinate system is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |= a 2 + b 2 . See [link].
  • In the rectangular coordinate system, unit vectors may be represented in terms of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>i</mi></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>j</mi></annotation-xml></semantics></math> where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the horizontal component and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the vertical component. Then, v = ai + bj  is a scalar multiple of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by real numbers<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mi>b</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>See [link] and [link].
  • Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j. See [link].
  • A vector v = ai + bj is written in terms of magnitude and direction as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |cos θi+| v |sin θj. See [link].
  • The dot product of two vectors is the product of the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>terms plus the product of the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>terms. See [link].
  • We can use the dot product to find the angle between two vectors. [link] and [link].
  • Dot products are useful for many types of physics applications. See [link].

Section Exercises

Verbal

What are the characteristics of the letters that are commonly used to represent vectors?

lowercase, bold letter, usually<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi></mrow></annotation-xml></semantics></math>

How is a vector more specific than a line segment?

What are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and what do they represent?

They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

What is component form?

When a unit vector is expressed as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩,which letter is the coefficient of the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and which the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>?</mo></mrow></annotation-xml></semantics></math>

The first number always represents the coefficient of the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and the second represents the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Algebraic

Given a vector with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,2 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,−3 ), find an equivalent vector whose initial point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ). Write the vector in component form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.

Given a vector with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −4,2 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,−3 ), find an equivalent vector whose initial point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ). Write the vector in component form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>〈</mo></mrow></mrow></annotation-xml></semantics></math> 7,−5 〉

Given a vector with initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,−1 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1,−7 ), find an equivalent vector whose initial point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ). Write the vector in component form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> a,b ⟩.

For the following exercises, determine whether the two vectors<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are equal, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has an initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1  and a terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2  and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math> has an initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 3  and a terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( 5,1 ), P 2 =( 3,−2 ), P 3 =( −1,3 ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( 9,−4 )

not equal

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( 2,−3 ), P 2 =( 5,1 ), P 3 =( 6,−1 ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( 9,3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( −1,−1 ), P 2 =( −4,5 ), P 3 =( −10,6 ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( −13,12 )

equal

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( 3,7 ), P 2 =( 2,1 ), P 3 =( 1,2 ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( −1,−4 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( 8,3 ), P 2 =( 6,5 ), P 3 =( 11,8 ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 4 =( 9,10 )

equal

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( −3,1 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( 5,2 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 6,0 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −1,−3 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>7</mn><mi>i</mi><mo>−</mo><mn>3</mn><mi>j</mi></mrow></annotation-xml></semantics></math>

For the following exercises, use the vectors u = i + 5j, v = −2i− 3j,  and w = 4ij.

Find u + (vw)

Find 4v + 2u

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>6</mn><mi>i</mi><mo>−</mo><mn>2</mn><mi>j</mi></mrow></annotation-xml></semantics></math>

For the following exercises, use the given vectors to compute u + v, uv, and 2u − 3v.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 2,−3 ⟩,v=⟨ 1,5 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,4 ⟩,v=⟨ −2,1 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo>+</mo><mi>v</mi><mo>=</mo><mrow><mo>〈</mo></mrow></mrow></annotation-xml></semantics></math> −5,5 〉,u−v=〈 −1,3 〉,2u−3v=〈 0,5 〉

Let v = −4i + 3j. Find a vector that is half the length and points in the same direction as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Let v = 5i + 2j. Find a vector that is twice the length and points in the opposite direction as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>10</mn><mi>i</mi><mo>–</mo><mn>4</mn><mi>j</mi></mrow></annotation-xml></semantics></math>

For the following exercises, find a unit vector in the same direction as the given vector.

a = 3i + 4j

b = −2i + 5j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 29 29 i+ 5 29 29 j

c = 10ij

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 i+ 5 2 j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 229 229 i+ 15 229 229 j

u = 100i + 200j

u = −14i + 2j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 2 10 i+ 2 10 j

For the following exercises, find the magnitude and direction of the vector,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo>≤</mo><mi>θ</mi><mo><</mo><mn>2</mn><mi>π</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 0,4 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 6,5 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=7.810,θ=39.806°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 2,−5 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −4,−6 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=7.211,θ=236.310°

Given u = 3i − 4j and v = −2i + 3j, calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Given u = −ij and v = i + 5j, calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>6</mn></mrow></annotation-xml></semantics></math>

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −2,4 ⟩ and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,1 ⟩, calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Given u<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −1,6 ⟩and v<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 6,−1 ⟩,calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>12</mn></mrow></annotation-xml></semantics></math>

Graphical

For the following exercises, given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>draw<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>,</mo></mrow></annotation-xml></semantics></math>3v and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2 v.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 2,−1 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −1,4 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,−2 ⟩

For the following exercises, use the vectors shown to sketch u + v, uv, and 2u.

Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,1) and v goes to (-1,2).
Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).
Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,2) and v goes to (1,-1).
Plot of vectors u and v located head to tail. Take u's start point as the origin. In terms of that, u goes from the origin to (3,-2), and v goes from (3,-2) to (2,-3)
Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).

For the following exercises, use the vectors shown to sketch 2u + v.

Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (3,1) and v goes from the origin to (2,-2).
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,-2) and v goes from the origin to (-3,-2).
Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).

For the following exercises, use the vectors shown to sketch u − 3v.

Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (-4,0) and v goes from the origin to (1,-1).
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,2) and v goes from the origin to (-2,1).
Vector extending from the origin to (7,5), taking the base as the origin.

For the following exercises, write the vector shown in component form.

Vector going from the origin to (-4,2).
Insert figure(table) alt text: Vector going from the origin to (4,1).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>〈</mo></mrow></mrow></annotation-xml></semantics></math> 4,1 〉

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 2,1 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −1,2 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then draw the vector on the graph.

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 4,−1 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −3,2 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Draw the points and the vector on the graph.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>v</mi><mo>=</mo><mo>−</mo><mn>7</mn><mi>i</mi><mo>+</mo><mn>3</mn><mi>j</mi></mrow></annotation-xml></semantics></math>

Vector going from (4,-1) to (-3,2).

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 3,3 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −3,3 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Draw the points and the vector on the graph.

Extensions

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=6,θ=45°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><msqrt/></mrow></annotation-xml></semantics></math> 2 i+3 2 j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=8,θ=220°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=2,θ=300°

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>i</mi><mo>−</mo><msqrt/></mrow></annotation-xml></semantics></math> 3 j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> v |=5,θ=135°

A 60-pound box is resting on a ramp that is inclined 12°. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.

a. 58.7; b. 12.5

A 25-pound box is resting on a ramp that is inclined 8°. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27° above the horizontal. Round to the nearest hundredth.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>7.13</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>pounds,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3.63</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>pounds

Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127° above the horizontal. Round to the nearest hundredth.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>2.87</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>pounds,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>4.10</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>pounds

Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8° above the horizontal. Round to the nearest hundredth.

Real-World Applications

A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

4.635 miles, 17.764° N of E

A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?

A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

17 miles. 10.318 miles

A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?

Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North

A woman starts walking from home and walks 6 miles at 40° north of east, then 2 miles at 15° east of south, then 5 miles at 30° south of west. If she walked straight home, how far would she have to walk, and in what direction?

An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

4.924°. 659 km/hr

An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing from the northwest at 50 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

4.424°

An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

As part of a video game, the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,7 ) is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of this point.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0.081,8.602 )

As part of a video game, the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,3 ) is rotated counterclockwise about the origin through an angle of 40°. Find the new coordinates of this point.

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

21.801°, relative to the car’s forward direction

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.

parallel: 16.28, perpendicular: 47.28 pounds

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45° from the horizontal. What single force is the resultant force acting on the body?

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135° from the horizontal, and 5 pounds acting on it directed 150° from the horizontal. What single force is the resultant force acting on the body?

19.35 pounds, 231.54° from the horizontal

The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45° from the horizontal. What single force is needed to produce a state of equilibrium on the body?

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

5.1583 pounds, 75.8° from the horizontal

Chapter Review Exercises

Non-right Triangles: Law of Sines

For the following exercises, assume<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>,</mo><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>γ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Solve each triangle, if possible. Round each answer to the nearest tenth.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>β</mi><mo>=</mo><mn>50°</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>105</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>45</mn></mrow></annotation-xml></semantics></math>

Not possible

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>α</mi><mo>=</mo><mn>43.1°</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>184.2</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>242.8</mn></mrow></annotation-xml></semantics></math>

Solve the triangle.

Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo>=</mo><mn>120°</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>23.1</mn><mo>,</mo><mi>c</mi><mo>=</mo><mn>34.1</mn></mrow></annotation-xml></semantics></math>

Find the area of the triangle.

A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in [link]. Find the distance of the plane from point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the elevation of the plane.

<figure class="small" id="Image_08_08_225">Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.</figure>

distance of the plane from point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>:</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>2.2 km, elevation of the plane: 1.6 km

Non-right Triangles: Law of Cosines

Solve the triangle, rounding to the nearest tenth, assuming<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>,</mo><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>γ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>s opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>c</mi><mo>:</mo><mtext> </mtext><mi>a</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>6</mn><mo>,</mo><mi>c</mi><mo>=</mo><mn>8.</mn></mrow></annotation-xml></semantics></math>

Solve the triangle in [link], rounding to the nearest tenth.

<figure class="small" id="Image_08_08_226">A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13. </figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>=</mo><mn>71.0°</mn><mo>,</mo><mi>C</mi><mo>=</mo><mn>55.0°</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>12.8</mn></mrow></annotation-xml></semantics></math>

Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

To find the distance between two cities, a satellite calculates the distances and angle shown in [link] (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

<figure class="small" id="Image_08_08_227">Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees. </figure>

40.6 km

Polar Coordinates

Plot the point with polar coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3, π 6 ).

Plot the point with polar coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5,− 2π 3 )

 

 
Polar coordinate grid with a point plotted on the fifth concentric circle 2/3 the way between pi and 3pi/2 (closer to 3pi/2).

 

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6,− 3π 4 ) to rectangular coordinates.

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2, 3π 2 ) to rectangular coordinates.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,2 ) 

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,−2 )to polar coordinates.

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −9,−4 ) to polar coordinates.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 9.8489,203.96° )

For the following exercises, convert the given Cartesian equation to a polar equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =64

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>8</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =−2y

For the following exercises, convert the given polar equation to a Cartesian equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>7</mn><mtext>cos</mtext><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =7x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −2 4cos θ+sin θ

For the following exercises, convert to rectangular form and graph.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3π 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mi>x</mi></mrow></annotation-xml></semantics></math>

Plot of the function y=-x in rectangular coordinates.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>5</mn><mi>sec</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math>

Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>4</mn><mo>+</mo><mn>4</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math>

symmetric with respect to the line<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>7</mn></mrow></annotation-xml></semantics></math>

Sketch a graph of the polar equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>5</mn><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Label the axis intercepts.

 

 
Graph of the given polar equation - an inner loop limaçon.

 

Sketch a graph of the polar equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>5</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7θ ).

Sketch a graph of the polar equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>3</mn><mo>−</mo><mn>3</mn><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math>

 

 
Graph of the given polar equation - a cardioid.

 

Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>+</mo><mn>6</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mo>−</mo><mtext>​</mtext><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

5

Write the complex number in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>+</mo><mn>9</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 − 3 2 ​i

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 3 )

For the following exercises, convert the complex number from polar to rectangular form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 40° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2.3</mn><mo>+</mo><mn>1.9</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

For the following exercises, find the product<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 z 2  in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =2cis( 89° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 2 =5cis( 23° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =10cis( π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 2 =6cis( π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 )

For the following exercises, find the quotient<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> z 1 z 2  in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =12cis( 55° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 2 =3cis( 18° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =27cis( 5π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 2 =9cis( π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4π 3 )

For the following exercises, find the powers of each complex number in polar form.

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 4  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 70° )

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 2  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 4 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>25</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 2 )

For the following exercises, evaluate each root.

Evaluate the cube root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>64</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 210° ).

Evaluate the square root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>25</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 4 ),5cis( 7π 4 )

For the following exercises, plot the complex number in the complex plane.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

Plot of -1 + 3i in the complex plane (-1 along the real axis, 3 along the imaginary).

Parametric Equations

For the following exercises, eliminate the parameter<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to rewrite the parametric equation as a Cartesian equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x( t )=3t−1 y( t )= t

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mtable columnalign="left"/></mrow></annotation-xml></semantics></math> x(t)=−cos t y(t)=2 sin 2 t 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + 1 2 y=1

Parameterize (write a parametric equation for) each Cartesian equation by using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=acos t and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mi>sin</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> x 2 25 + y 2 16 =1.

Parameterize the line from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>7</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>so that the line is at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>7</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x( t )=−2+6t y( t )=3+4t

Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x( t )=3 t 2 y( t )=2t−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x(t)= e t y(t)=−2 e 5 t

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 5

 

 

Plot of the given parametric equations.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x(t)=3cos t y(t)=2sin t

A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 3 seconds?
  3. How long is the ball in the air?
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x( t )=( 80cos( 40° ) )t y( t )=−16 t 2 +( 80sin( 40° ) )t+4
  2. The ball is 14 feet high and 184 feet from where it was launched.
  3. 3.3 seconds

Vectors

For the following exercises, determine whether the two vectors,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>are equal, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has an initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1  and a terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has an initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 3  and a terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( −1,4 ), P 2 =( 3,1 ), P 3 =( 5,5 )and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( 9,2 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>P</mi></msub></mrow></annotation-xml></semantics></math> 1 =( 6,11 ), P 2 =( −2,8 ), P 3 =( 0,−1 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 4 =( −8,2 )

not equal

For the following exercises, use the vectors<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mstyle mathsize="normal" mathvariant="bold"><mo>=</mo></mstyle><mn>2</mn><mi>i</mi><mo>−</mo><mi>j</mi><mtext>,</mtext><mi>v</mi><mo>=</mo><mn>4</mn><mi>i</mi><mo>−</mo><mn>3</mn><mi>j</mi><mtext>,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>w</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>i</mi><mo>+</mo><mn>5</mn><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to evaluate the expression.

uv

2vu + w

4i

For the following exercises, find a unit vector in the same direction as the given vector.

a = 8i − 6j

b = −3ij

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 10 10i<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 10 10j

For the following exercises, find the magnitude and direction of the vector.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> 6,−2 ⟩

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>⟨</mo></mrow></mrow></annotation-xml></semantics></math> −3,−3 ⟩

Magnitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><msqrt/></mrow></annotation-xml></semantics></math> 2 , Direction:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>225°</mtext></mrow></annotation-xml></semantics></math>

For the following exercises, calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mtext>.</mtext></mrow></annotation-xml></semantics></math>

u = −2i + j and v = 3i + 7j

u = i + 4j and v = 4i + 3j

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>16</mtext></mrow></annotation-xml></semantics></math>

Given v<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>〈</mo></mrow></mrow></annotation-xml></semantics></math> −3,4 〉draw v, 2v, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2v.

Given the vectors shown in [link], sketch u + v, uv and 3v.

<figure class="small" id="Image_08_08_241">Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude. </figure>

 

 
Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

 

Given initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 3,2 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −5,−1 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mi>i</mi><mtext> </mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Draw the points and the vector on the graph.

Practice Test

Assume<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>,</mo><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>γ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is opposite side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Solve the triangle, if possible, and round each answer to the nearest tenth, given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mo>=</mo><mn>68°</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>21</mn><mo>,</mo><mi>c</mi><mo>=</mo><mn>16.</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>α</mi><mo>=</mo><mn>67.1°</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn>44.9°</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>20.9</mn></mrow></annotation-xml></semantics></math>

Find the area of the triangle in [link]. Round each answer to the nearest tenth.

<figure class="small" id="Image_08_08_244">A triangle. One angle is 60 degrees with opposite side 6.25. The other two sides are 5 and 7. </figure>

A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>1712 miles</mtext></mrow></annotation-xml></semantics></math>

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2,2 )  to polar coordinates, and then plot the point.

Convert<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2, π 3 ) to rectangular coordinates.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1, 3 )

Convert the polar equation to a Cartesian equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> x 2 + y 2 =5y.

Convert to rectangular form and graph:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>csc</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics></math>

 

 

Plot of the given equation in rectangular form - line y=-3.

Test the equation for symmetry:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2θ ).

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

 

 
Graph of the given equations - a cardioid.

 

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>3</mn><mo>−</mo><mn>5</mn><mtext>sin</mtext><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Find the absolute value of the complex number <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>−</mo><mn>9</mn><mi>i</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mrow><mn>106</mn></mrow></msqrt></mrow></annotation-xml></semantics></math>

Write the complex number in polar form:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 4+i.

Convert the complex number from polar to rectangular form:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>5</mn><mtext>cis</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2π 3 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 +i 5 3 2

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 =8cis( 36° ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 =2cis( 15° ),evaluate each expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 z 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msub><mi>z</mi></msub></mrow></mfrac></mrow></annotation-xml></semantics></math> 1 z 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mi>cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 21° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> z 2 ) 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mrow><msub><mi>z</mi></msub></mrow></msqrt></mrow></annotation-xml></semantics></math> 1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msqrt/></mrow></annotation-xml></semantics></math> 2 cis( 18° ),2 2 cis( 198° )

Plot the complex number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−5</mn><mo>−</mo><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the complex plane.

Eliminate the parameter<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to rewrite the following parametric equations as a Cartesian equation: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x(t)=t+1 y(t)=2 t 2 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> ( x−1 ) 2

Parameterize (write a parametric equation for) the following Cartesian equation by using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=acos t and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>:</mo></mrow></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 36 + y 2 100 =1.

Graph the set of parametric equations and find the Cartesian equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x(t)=−2sin t y(t)=5cos t .

 

 
Graph of the given equations - a vertical ellipse.

 

A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal. The ball is released at a height of 3.5 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 2 seconds?
  3. How long is the ball in the air?

For the following exercises, use the vectors u = i − 3j and v = 2i + 3j.

Find 2u − 3v.

−4i − 15j

Calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo>⋅</mo><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Find a unit vector in the same direction as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 13 13 i+ 3 13 13 j

Given vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has an initial point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 1 =( 2,2 ) and terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> P 2 =( −1,0 ), write the vector<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>On the graph, draw<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>v</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mi>v</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Glossary

dot product
given two vectors, the sum of the product of the horizontal components and the product of the vertical components
initial point
the origin of a vector
magnitude
the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem
resultant
a vector that results from addition or subtraction of two vectors, or from scalar multiplication
scalar
a quantity associated with magnitude but not direction; a constant
scalar multiplication
the product of a constant and each component of a vector
standard position
the placement of a vector with the initial point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ) and the terminal point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>represented by the change in the x-coordinates and the change in the y-coordinates of the original vector
terminal point
the end point of a vector, usually represented by an arrow indicating its direction
unit vector
a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> v 1 =⟨ 1,0 ⟩ the vertical unit vector runs along the y-axis and is defined as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> v 2 =⟨ 0,1 ⟩.
vector
a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)
vector addition
the sum of two vectors, found by adding corresponding components