11.4.2: Homework
- Page ID
- 197633
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reading Questions
- What two properties define a vector?
- What is the difference between a vector and a scalar?
- In a graphical representation of a vector, what does the length of the arrow represent? What does the arrowhead represent?
- When are two vectors considered equal?
- What is the term for the length of a vector?
- Describe what happens to a vector when it is multiplied by a positive scalar \(k > 1\).
- Describe what happens to a vector when it is multiplied by a negative scalar, such as -1.
- What is a resultant vector?
- Explain the "head-to-tail" method for adding two vectors.
- What is the "parallelogram rule" for vector addition?
- Is the magnitude of the sum of two vectors, \(||\vec{u} + \vec{v}||\), typically equal to the sum of their individual magnitudes, \(||\vec{u}|| + ||\vec{v}||\)? Why or why not?
Skills Refresher
Review the following skills you will need for this section.
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State two versions of the Law of Sines.
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State two versions of the Law of Cosines.
For the following exercises, find the unknown part of the triangle. Round to two decimal places.
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\(A=15^{\circ}\), \(B=125^{\circ}\), and \(b=12\) cm. Find \(a\).
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\(C=87^{\circ}\), \(b=11\) inches, and \(c=13\) inches. Find \(B\).
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\(A=37^{\circ}\), \(b=6\), and \(c=14\). Find \(a\).
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\(a=9\), \(b=4\), and \(c=7\). Find \(B\).
- Answers
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\(\dfrac{\sin\left(A\right)}{a} = \dfrac{\sin\left(B\right)}{b}, \dfrac{b}{\sin\left(B\right)} = \dfrac{c}{\sin\left(C\right)}\)
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\(a^2=b^2+c^2-2bc \cos \left(A\right), b^2 = a^2 + c^2 -2ac \cos \left(B\right)\)
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\(3.79\)
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\(57.67^{\circ}\)
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\(9.89\)
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\(25.21^{\circ}\)
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Homework
Vocabulary Check
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A quantity defined by both a magnitude (such as a distance) and a direction is called a ___.
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Two vectors are equal if they have the same ___ and ___; it does not matter where the vector starts.
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A quantity having only magnitude (and no direction) is called a ___.
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The length of a vector \(\vec{v}\) is called its ___, and is denoted by \(\norm{\vec{v}}\).
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The sum of two vectors \(\vec{u}\) and \(\vec{v}\) is a new vector, \(\vec{w}\), starting at the ___ of the first vector and ending at the ___ of the second vector.
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The sum of two vectors is called the ___ vector.
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The rule for adding vectors is sometimes called the ___ rule.
Concept Check
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What is the difference between a scalar and a vector?
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If velocity is represented by a vector, what is its magnitude called?
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Does \(\norm{\vec{u}+\vec{v}} = \norm{\vec{u}} + \norm{\vec{v}}\)?
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Does \(\norm{k \vec{v}} = |k| \cdot \norm{\vec{v}}\)?
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What is the parallelogram rule?
True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.
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If \(k \lt 0\), the magnitude of \(k \vec{v}\) is \(k\) times the magnitude of \(\vec{v}\).
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The direction of \(k \vec{v}\) is the same as the direction of \(\vec{v}\).
Basic Skills
For the following exercises, sketch a vector to represent the quantity.
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The waterfall is 3 km away, at a bearing of \(\mathrm{S} \, 75^{\circ} \, \mathrm{W}\).
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The cave entrance is 450 meters away, at a bearing of \(\mathrm{N} \, 45^{\circ} \, \mathrm{E}\).
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The current is moving 6 feet per second at a heading of \(60^{\circ}\).
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The bird is flying due south at 45 miles per hour.
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The projectile was launched at a speed of 40 meters per second, at an angle of \(30^{\circ}\) above horizontal.
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The baseball was hit straight up at a speed of 60 miles per hour.
For the following exercises, which vectors are equal?
For the following exercises, sketch a vector equal to \( \vec{v} \), but starting at the given point.
For the following exercises, draw the scalar multiples of the given vectors.
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\(-2v \text{ and } 1.5v\) -
\(-\dfrac{1}{2}w \text{ and } 3w\) -
\(-2.5u \text{ and } \sqrt{2}u\) -
\(-\sqrt{6}t \text{ and } 5.4t\)
For the following exercises,
(a) draw the resultant vector,
(b) calculate the length and direction of the resultant vector.
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\(\vec{A} = \vec{u} + \vec{v}\)
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\(\vec{B} = \vec{z} + \vec{u}\)
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\(\vec{C} = \vec{w} + \vec{u}\)
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\(\vec{D} = \vec{G} + \vec{z}\)
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\(\vec{E} = \vec{z} + \vec{F}\)
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\(\vec{F} = \vec{w} + \vec{v}\)
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\(\vec{G} = \vec{w} + \vec{w}\)
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\(\vec{H} = \vec{G} + \vec{G}\)
For the following exercises, find the magnitude and direction of the vector.
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\(v_x = 5\), \(v_y = -12\)
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\(v_x = -8\), \(v_y = 15\)
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\(v_x = -6\), \(v_y = -7\)
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\(v_x = 1\), \(v_y = -3\)
For the following exercises, sketch the vectors, then calculate the resultant.
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Add the vector \(\vec{v}\) of length 45 pointing \(26^{\circ}\) east of north to the vector \(\vec{w}\) of length 32 pointing \(17^{\circ}\) south of west.
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Add the vector \(\vec{v}\) of length 105 pointing \(41^{\circ}\) west of south to the vector w of length 77 pointing \(8^{\circ}\) west of north.
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Let \(\vec{v}\) have length 8 and point in the heading \(10^{\circ}\). Let \(\vec{w}\) have length 13 and point in the heading \(250^{\circ}\). Find \(\vec{v}+\vec{w}\).
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Let a have length 43 and point in the heading \(343^{\circ}\). Let \(\vec{b}\) have length 19 and point in the heading \(141^{\circ}\). Find \(\vec{a}+\vec{b}\).
Synthesis Questions
Multiplying a vector \(\vec{v}\) by \(-1\) gives a vector \(-\vec{v}\) that has the same magnitude as \(\vec{v}\) but points in the opposite direction. We define subtraction of two vectors the same way we define subtraction of integers:\[\vec{u}-\vec{v}=\vec{u}+(-\vec{v}). \nonumber \]That is, to subtract a vector \(\vec{v}\), we add its opposite.
For the following exercises, draw the resultant vector.
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\(\vec{A}=\vec{u}-\vec{v}\)
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\(\vec{B}=\vec{F}-\vec{z}\)
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\(\vec{C}=\vec{v}-\vec{u}\)
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\(\vec{D}=\vec{z}-\vec{G}\)
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\(\vec{P}=\vec{w}-\vec{F}\)
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\(\vec{Q}=\vec{u}-\vec{w}\)
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\(\vec{R}=\vec{G}-\vec{u}\)
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\(\vec{S}=\vec{v}-\vec{F}\)
Applications
For the following exercises, sketch the vectors, then calculate the resultant.
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Loi swam 3.6 miles at a bearing of \(\mathrm{N} \, 20^{\circ} \, \mathrm{E}\). However, the water current displaced her by 0.9 miles at a bearing of \(\mathrm{N} \, 37^{\circ} \, \mathrm{E}\). How far is Loi from her starting point, and at what bearing?
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Paige paddles her canoe 4.5 miles at a bearing of \(\mathrm{N} \, 12^{\circ} \, \mathrm{W}\). The water current pushes her 0.3 miles off course at a bearing of \(\mathrm{N} \, 5^{\circ} \, \mathrm{E}\). How far is Paige from her starting point, and at what bearing?
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Nhat wants to fly to an airport that is 103 miles due west in 1 hour. The prevailing winds blow at a heading of \(112^{\circ}\) at 28 miles per hour, so Nhat will head her plane somewhat north of due west to compensate. What airspeed and heading should Nhat take?
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Nam wants to cross a 300 meter wide river, but the river is running due south at 80 meters per minute. There are rocks upstream and rapids downstream, so he wants to paddle straight across from east to west. At what heading should he point his kayak, and how fast should his water speed be in order to cross the river in 2 minutes? (Hint: The current will move him 160 meters due south compared with where his speed and heading would take him if the current stopped. Compute the distance he would have traveled, then divide by 2 minutes to get the speed.)
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A ship maintains a heading of \(30^{\circ}\) and a speed of 20 miles per hour. There is a current in the water running at a heading of \(135^{\circ}\) and at a speed of 10 miles per hour. What is the actual heading and speed of the ship?
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A plane is heading due south, with an airspeed of 180 kilometers per hour. The wind is blowing at 50 kilometers per hour at a heading of \(225^{\circ}\). What is the actual heading and speed of the plane?
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The campground is 3.6 kilometers from the trail head at a bearing of \(\mathrm{N} \, 20^{\circ} \, \mathrm{W}\). A ranger station is located 2.3 kilometers from the campsite at a bearing of \(\mathrm{S} \, 8^{\circ} \, \mathrm{W}\). What is the distance and bearing from the trail head to the ranger station?
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There is treasure buried 40 paces due east from a dead tree. From the buried treasure, a hidden mine shaft is 100 paces at a bearing of \(\mathrm{N} \, 58^{\circ} \, \mathrm{W}\). What is the distance and bearing from the dead tree to the mine shaft?