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Mathematics LibreTexts

Section 2:

Skills to Develop

In this section, we strive to understand the ideas generated by the following important questions:

  • What is a vector?
  • What does it mean for two vectors to be equal?
  • How do we add two vectors together and multiply a vector by a scalar?
  • How do we determine the magnitude of a vector? What is a unit vector, and how do we find a unit vector in the direction of a given vector?

Introduction

If we are at a point \(x\) in the domain of a function of one variable, there are only two directions in which we can move: in the positive or negative x-direction. If, however, we are at a point \((x, y)\) in the domain of a function of two variables, there are many directions in which we can move. Thus, it is important for us to have a means to indicate direction, and we will do so using vectors.

Preview Activity \(\PageIndex{1}\):

After working out, Sarah and John leave the Recreation Center on the Grand Valley State University Allendale campus (a map of which is given in Figure 9.20) to go to their next classes.4 Suppose we record Sarah’s movement on the map in a pair \(<x; y>\) (we will call this pair a vector), where \(x\) is the horizontal distance (in feet) she moves (with east as the positive direction) and \(y\) as the vertical distance (in feet) she moves (with north as the positive direction). We do the same for John. Throughout, use the legend to estimate your responses as best you can.

(a) What is the vector \(v_1 = <x, y>\) that describes Sarah’s movement if she walks directly in a straight line path from the Recreation Center to the entrance at the northwest end of Mackinac Hall? (Assume a straight line path, even if there are buildings in the way.) Explain how you found this vector. What is the total distance in feet between the Recreation Center and the entrance to Mackinac Hall? Measure the number of feet directly and then explain how to calculate this distance in terms of \(x\) and \(y\).

(b) What is the vector \(v_2 = <x, y>\) that describes John’s change in position if he walks directly from the Recreation Center to Au Sable Hall? How many feet are there between Recreation Center to Au Sable Hall in terms of \(x\) and \(y\)?

(c) What is the vector \(v_3 = <x, y>\) that describes the change in position if John walks directly from Au Sable Hall to the northwest entrance of Mackinac Hall to meet up with Sarah after class? What relationship do you see among the vectors \(v_1\), \(v_2\), and \(v_3\)? Explain why this relationship should hold.

 

Figure 9.20: Grand Valley State University Allendale campus map.

4: 4GVSU campus map from http://www.gvsu.edu/homepage/files/p.../allendale.pdf, used with permission from GVSU, credit to illustrator Chris Bessert.

Representations of Vectors

Preview Activity \(\PageIndex{1}\) shows how we can record the magnitude and direction of a change in position using an ordered pair of numbers \(<x, y>\). There are many other quantities, such as force and velocity, that possess the attributes of magnitude and direction, and we will call each such quantity a vector.

Definition: Vector

A vector is any quantity that possesses the attributes of magnitude and direction.

We can represent a vector geometrically as a directed line segment, with the magnitude as the length of the segment and an arrowhead indicating direction, as shown in Figure 9.21.

 

Figure 9.21: A vector.

 

Figure 9.22: Representations of the same vector.

According to the definition, a vector possesses the attributes of length (magnitude) and direction; the vector’s position, however, is not mentioned. Consequently, we regard as equal any two vectors having the same magnitude and direction, as shown in Figure 9.22.

Note

Two vectors are equal provided they have the same magnitude and direction.

This means that the same vector may be drawn in the plane in many different ways. For instance, suppose that we would like to draw the vector \(<3, 4>\), which represents a horizontal change of three units and a vertical change of four units. We may place the tail of the vector (the point from which the vector originates) at the origin and the tip (the terminal point of the vector) at (3; 4), as illustrated in Figure 9.23. A vector with its tail at the origin is said to be in standard position.

Alternatively, we may place the tail of the vector \(<3, 4>\) at another point, such as \(Q(1, 1)\). After a displacement of three units to the right and four units up, the tip of the vector is at the point \(R(4, 5)\) (see Figure 9.24).

 

Figure 9.23: A vector in standard position


Figure 9.24: A vector between two points

In this example, the vector led to the directed line segment from \(Q\) to \(R\), which we denote as \(\vec{QR}\). We may also turn the situation around: given the two points \(Q\) and \(R\), we obtain the vector \(<3, 4>\) because we move horizontally three units and vertically four units to get from \(Q\) to \(R\). In other words, \(\vec{QR} = <3,4>\). In general, the vector \(\vec{QR}\) from the point \(Q=(q_1,q_2)\) to \(R = (r_1, r_2)\) is found by taking the difference of coordinates, so that

��Q��!R = hr1 �� q1; r2 �� q2i:

We will use boldface letters to represent vectors, such as \(v = <3, 4>\), to distinguish them from scalars. The entries of a vector are called its components; in the vector \(<3, 4>\), the \(x\) component is 3 and the \(y\) component is 4. We use pointed brackets \(<\,,\,>\) and the term components to distinguish a vector from a point \((\, , \,)\) and its coordinates. There is, however, a close connection between vectors and points. Given a point \(P\), we will frequently consider the vector \(\vec{OP}\) from the origin \(O\) to \(P\). For instance, if \(P = (3, 4)\), then \(\vec{OP}=<3,4>\) as in Figure 9.25. In this way, we think of a point \(P\) as defining a vector \(\vec{OP}\) whose components agree with the coordinates of \(P\).


Figure 9.25: A point defines a vector

While we often illustrate vectors in the plane since it is easier to draw pictures, different situations call for the use of vectors in three or more dimensions. For instance, a vector \(v\) in n-dimensional space, \(\mathbb{R}^n\), has \(n\) components and may be represented as

v = hv1; v2; v3; : : : ; vni:

The next activity will help us to become accustomed to vectors and operations on vectors in three dimensions.

Activity \(\PageIndex{1}\):

As a class, determine a coordinatization of your classroom, agreeing on some convenient set of axes (e.g., an intersection of walls and floor) and some units in the \(x\), \(y\), and \(z\) directions (e.g., using lengths of sides of floor, ceiling, or wall tiles). Let \(O\) be the origin of your coordinate system. Then, choose three points, \(A\), \(B\), and \(C\) in the room, and complete the following.

(a) Determine the coordinates of the points \(A\), \(B\), and \(C\).
(b) Determine the components of the indicated vectors.
(i) O��!A (ii) O����!B (iii) O����!C (iv) A����!B (v) A��!C (vi) B����!C
C

Equality of Vectors

Because location is not mentioned in the definition of a vector, any two vectors that have the same magnitude and direction are equal. It is helpful to have an algebraic way to determine when this occurs. That is, if we know the components of two vectors u and v, we will want to be able to determine algebraically when u and v are equal. There is an obvious set of conditions that we use.

Note

Two vectors u = hu1; u2i and v = hv1; v2i in R2 are equal if and only if their corresponding components are equal: u1 = v1 and u2 = v2. More generally, two vectors u = hu1; u2; : : : ; uni and v = hv1; v2; : : : ; vni in Rn are equal if and only if ui = vi for each possible value of i.

Operations on Vectors

Vectors are not numbers, but we can now represent them with components that are real numbers. As such, we naturally wonder if it is possible to add two vectors together, multiply two vectors, or combine vectors in any other ways. In this section, we will study two operations on vectors:
vector addition and scalar multiplication. To begin, we investigate a natural way to add two vectors together, as well as to multiply a vector by a scalar.

Activity \(\PageIndex{1}\):

Let u = h2; 3i, v = h��1; 4i.
(a) Using the two specific vectors above, what is the natural way to define the vector sum
u + v?
(b) In general, how do you think the vector sum a+b of vectors a = ha1; a2i and b = hb1; b2i in R2 should be defined? Write a formal definition of a vector sum based on your
intuition.
(c) In general, how do you think the vector sum a + b of vectors a = ha1; a2; a3i and
b = hb1; b2; b3i in R3 should be defined? Write a formal definition of a vector sum
based on your intuition.
(d) Returning to the specific vector v = h��1; 4i given above, what is the natural way to
define the scalar multiple 1
2v?
(e) In general, how do you think a scalar multiple of a vector a = ha1; a2i in R2 by a scalar
c should be defined? how about for a scalar multiple of a vector a = ha1; a2; a3i in R3
by a scalar c? Write a formal definition of a scalar multiple of a vector based on your
intuition.

We can now add vectors and multiply vectors by scalars, and thus we can add together scalar multiples of vectors. This allows us to define vector subtraction, v��u, as the sum of v and ��1 times u, so that

v �� u = v + (��1)u:

Using vector addition and scalar multiplication, we will often represent vectors in terms of the special vectors i = h1; 0i and j = h0; 1i. For instance, we can write the vector ha; bi in R2 as

ha; bi = ah1; 0i + bh0; 1i = ai + bj;

which means that

h2;��3i = 2i �� 3j:

In the context of R3, we let i = h1; 0; 0i, j = h0; 1; 0i, and k = h0; 0; 1i, and we can write the vector ha; b; ci in R3 as

ha; b; ci = ah1; 0; 0i + bh0; 1; 0i + ch0; 0; 1i = ai + bj + ck:

The vectors i, j, and k are called the standard unit vectors5, and are important in the physical sciences.

Properties of Vector Operations

We know that the scalar sum 1 + 2 is equal to the scalar sum 2 + 1. This is called the commutative property of scalar addition. Any time we define operations on objects (like addition of vectors) we usually want to know what kinds of properties the operations have. For example, is addition of vectors a commutative operation? To answer this question we take two arbitrary vectors v and u 5As we will learn momentarily, unit vectors have length 1 and add them together and see what happens. Let v = hv1; v2i and u = hu1; u2i. Now we use the fact that v1, v2, u1, and u2 are scalars, and that the addition of scalars is commutative to see that

v + u = hv1; v2i + hu1; u2i = hv1 + u1; v2 + u2i = hu1 + v1; u2 + v2i = hu1; u2i + hv1; v2i = u + v:

So the vector sum is a commutative operation. Similar arguments can be used to show the following properties of vector addition and scalar multiplication.

Theorem

Let v, u, and w be vectors in Rn and let a and b be scalars. Then
1. v + u = u + v
2. (v + u) + w = v + (u + w)
3. The vector 0 = h0; 0; : : : ; 0i has the property that v + 0 = v. The vector 0 is called the
zero vector.
4. (��1)v + v = 0. The vector (��1)v = ��v is called the additive inverse of the vector v.
5. (a + b)v = av + bv
6. a(v + u) = av + au
7. (ab)v = a(bv)
8. 1v = v.

We verified the first property for vectors in R2; it is straightforward to verify that the rest of the eight properties just noted hold for all vectors in Rn.

Geometric Interpretation of Vector Operations

Next, we explore a geometric interpretation of vector addition and scalar multiplication that allows us to visualize these operations. Let u = h4; 6i and v = h3;��2i. Then w = u + v = h7; 4i, as shown on the left in Figure 9.26.

If we think of these vectors as displacements in the plane, we find a geometric way to envision vector addition. For instance, the vector u+v will represent the displacement obtained by following the displacement u with the displacement v. We may picture this by placing the tail of v at the
tip of u, as seen in the center of Figure 9.26.

Of course, vector addition is commutative so we obtain the same sum if we place the tail of u at the tip of v. We therefore see that u+v appears as the diagonal of the parallelogram determined by u and v, as shown on the right of Figure 9.26.

Vector subtraction has a similar interpretation. On the left in Figure 9.27, we see vectors u, v, and w = u + v. If we rewrite v = w �� u, we have the arrangement of Figure 9.28. In other words, to form the difference w �� u, we draw a vector from the tip of u to the tip of w.

Figure 9.26: A vector sum (left), summing displacements (center), the parallelogram law (right)

Figure 9.27: Vector addition

Figure 9.28: Vector subtraction

In a similar way, we may geometrically represent a scalar multiple of a vector. For instance, if v = h2; 3i, then 2v = h4; 6i. As shown in Figure 9.29, multiplying v by 2 leaves the direction unchanged, but stretches v by 2. Also, ��2v = h��4;��6i, which shows that multiplying by a
negative scalar gives a vector pointing in the opposite direction of v.

Figure 9.29: Scalar multiplication

Activity \(\PageIndex{1}\):

Figure 9.30

Figure 9.31

Suppose that u and v are the vectors shown in Figure 9.30.
(a) On Figure 9.30, sketch the vectors u + v, v �� u, 2u, ��2u, and ��3v.
(b) What is 0v?
(c) On Figure 9.31, sketch the vectors ��3v, ��2v, ��1v, 2v, and 3v.
(d) Give a geometric description of the set of vectors tv where t is any scalar.
(e) On Figure 9.31, sketch the vectors u �� 3v, u �� 2v, u �� v, u + v, and u + 2v.
(f) Give a geometric description of the set of vectors u + tv where t is any scalar.

The Magnitude of a Vector

By definition, vectors have both direction and magnitude (or length). We now investigate how to calculate the magnitude of a vector. Since a vector v can be represented by a directed line segment, we can use the distance formula to calculate the length of the segment. This length is the magnitude of the vector v and is denoted jvj.

Activity \(\PageIndex{1}\):

 

Figure 9.32: The vector defined by A and B.

 

Figure 9.33: An arbitrary vector, v.

(a) Let A = (2; 3) and B = (4; 7), as shown in Figure 9.32. Compute jA����!Bj.
(b) Let v = hv1; v2i be the vector in R2 with components v1 and v2 as shown in Figure 9.33.
Use the distance formula to find a general formula for jvj.
(c) Let v = hv1; v2; v3i be a vector in R3. Use the distance formula to find a general formula
for jvj.
(d) Suppose that u = h2; 3i and v = h��1; 2i. Find juj, jvj, and ju + vj. Is it true that
ju + vj = juj + jvj?
(e) Under what conditions will ju+vj = juj+jvj? (Hint: Think about how u, v, and u+v
form the sides of a triangle.)
(f) With the vector u = h2; 3i, find the lengths of 2u, 3u, and ��2u, respectively, and use
proper notation to label your results.
(g) If t is any scalar, how is jtuj related to juj?
(h) A unit vector is a vector whose magnitude is 1. Of the vectors i, j, and i + j, which are
unit vectors?
(i) Find a unit vector v whose direction is the same as u = h2; 3i. (Hint: Consider the result
of part (g).)

Summary

  • A vector is any object that possesses the attributes of magnitude and direction. Examples of vector quantities are position, velocity, acceleration, and force.
  • Two vectors are equal if they have the same direction and magnitude. Notice that position is not considered, so a vector is independent of its location.
  • If u = hu1; u2; : : : ; uni and v = hv1; v2; : : : ; vni are two vectors in Rn, then their vector sum is the vector

u + v = hu1 + v1; u2 + v2; : : : ; un + vni:

  • If u = hu1; u2; : : : ; uni is a vector in Rn and c is a scalar, then the scalar multiple cu is the vector

cu = hcu1; cu2; : : : ; cuni:

  • The magnitude of the vector v = hv1; v2; : : : ; vni in Rn is the scalar

 

A vector u is a unit vector provided that juj = 1. If v is a nonzero vector, then the vector v jvj is a unit vector with the same direction as v.