**9.2**

**Vectors**

**Motivating Questions**

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 pointxin the domain of a function of one variable, there are only two directions in which we can move: in the positive or negativex-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 9.2.** 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 Figure9.20) to go to their
next classes.4 Suppose we record Sarah’s movement on the map in a pairhx, y_{i}(we will call this
pair a vector), where x is the horizontal distance (in feet) she moves (with east as the positive
direction) andyas 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 vectorv1 = hx, yithat 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 Mack- inac 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 ofxandy.

(b) What is the vectorv2 =hx, yithat 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 ofxandy?

(c) What is the vectorv3 = hx, yithat 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

4_{GVSU campus map from}_{http://www.gvsu.edu/homepage/files/pdf/maps/allendale.pdf}_{, used with}

class? What relationship do you see among the vectorsv1, v2, andv3? Explain why this

relationship should hold.

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1 Alumni House & Visitor Center (AH)...E1 2 Au Sable Hall (ASH)... F5 3 Alexander Calder Fine Arts Center (CAC)... G6 4 Art Gallery Support Building (AGS)...E8 5 Calder Residence (CR)... G6 6 Campus Health Center (UHC)...E8 7 Central Utilities Building... F1 8 Children’s Enrichment Center (CC)...C5 9 The Commons (COM)... F4 10 The Connection (CON)...E7 11 Cook Carillon Tower...E5 12 Cook–DeWitt Center (CDC)...E5 13 James M. Copeland Living Center (COP)... F3 14 Richard M. DeVos Living Center (DLC)... G2 15 Fieldhouse (FH)...E3 15a Recreation Center (RC)...E4 16 Football Center (FC)...C2 17 Edward J. Frey Living Center (FLC)... G2 Grand Valley Apartments (GVA): 18 Benzie...F9 19 Keweenaw...F9 20 Mackinac ...F9 21 Oakland...F8 22 Office ...F9 23 Tuscola ...F8 24 Wexford ... F8 25 Grand Valley State University Arboretum... F5 26 Great Lakes Plaza... F5 27 Henry Hall (HRY)...E4 28 Arthur C. Hills Living Center (HLC)... G3 29 Icie Macy Hoobler Living Center (HLL)... G4 30 Paul A. Johnson Living Center (JLC)... G4 31 Kelly Family Sports Center (KTB)... D3 32 Russel H. Kirkhof Center (KC)...E5 33 William A. Kirkpatrick Living Center (KRP)... G3 34 Robert Kleiner Commons (KLC)... F2 35 Grace Olsen Kistler Living Center (KIS)... F3 36 Lake Huron Hall (LHH)... F5-6 37 Lake Michigan Hall (LHM)... F6 38 Lake Ontario Hall (LOH)... F6 39 Lake Superior Hall (LSH)... F6 40 Laker Village Apartments (LVA)... D-E7 41 Loutit Lecture Halls (LTT)... F4

42 Arend D. Lubbers Stadium...C2 43 Mackinac Hall (MAK)... F3 44 Manitou Hall (MAN)... F4 45 Meadows Club House (MCH)... A5 46 Meadows Learning Center (MLC)... A5 47 Multi-Purpose Facility (MPF)...C2 48 Mark A. Murray Living Center (MUR)... F7 49 Glenn A. Niemeyer East Living Center (NMR). F-G7 50 Glenn A. Niemeyer Honors Hall (HON)... F7 51 Glenn A. Niemeyer West Living Center (NMR).... F7 52 North Living Center A (NLA)... F2 53 North Living Center B (NLB)... G2 54 North Living Center C (NLC)... G2 55 Arnold C. Ott Living Center (OLC)... G4 56 Outdoor Recreation & Athletic Fields... B-C5 57 Seymour & Esther Padnos Hall of Science (PAD). F4 58 Performing Arts Center (PAC)... E-F6 59 Mary Idema Pew Library Learning and

Information Commons (LIB)...E5 60 Robert C. Pew Living Center (PLC)... G3 61 William F. Pickard Living Center (PKC)... G3 67 Public Safety (in Service Building)...E2 62 Ravine Apartments (RA)... D2 63 Ravine Center (RC)... D2 64 Kenneth W. Robinson Living Center (ROB)... F3 65 Seidman House (SH)... F6 66 Bill & Sally Seidman Living Center (SLC)... G3 67 Service Building (SER)...E2 68 South Apartment C (SAC)... F8 69 South Apartment D (SAD)... F9 70 South Apartment E (SAE)...E9 71 South Utilities Building (SUB)... G7 72 Dale Stafford Living Center (STA)... H3 73 Student Services Building (STU)... F4-5 74 Maxine M. Swanson Living Center (SWN)... H3 75 Ronald F. VanSteeland Living Center (VLC)... F7 76 Ella Koeze Weed Living Center (WLC)... F4 77 West Living Center A (WLA)... F3 78 West Living Center B (WLB)... F2 79 James H. Zumberge Hall (JHZ)... F5

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*Grand River and*
*Grand Valley*
*Boathouse*
*Little Mac*
*Bridge*
**Map Legend**
University Buildings
Parking Lots
State Highway
Main Roads
Minor Campus Roads
Pathways/Sidewalks
Arboretum (with Walkway)

Tennis
Courts
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*Map by Christopher J. Bessert • 10/2013 (Rev. 21)*

*www.chrisbessert.org (616) 878-4285*
*©2005, 2013 Grand Valley State University*

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** CALD**
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denotes areas
of campus
under construction

**Allendale Campus**

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9.2. VECTORS 21

**Representations of Vectors**

Preview Activity9.2 shows how we can record the magnitude and direction of a change in po- sition using an ordered pair of numbers hx, yi. 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 avector.

**Definition 9.6.** A**vector**is any quantity that possesses the attributes of magnitude and direc-
tion.

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 Figure9.21.

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 direc- tion; 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 Figure9.22.

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 h3,4i, which represents a horizontal change of three units and a vertical change of four units. We may place thetailof the vector (the point from which the vector originates) at the origin and thetip(the terminal point of the vector) at(3,4), as illustrated in Figure9.23. A vector with its tail at the origin is said to be instandard position.

Alternatively, we may place the tail of the vectorh3,4iat another point, such asQ(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 Figure9.24).

3 4

x y

Figure 9.23: A vector in standard position

1 4 1 5 x y Q(1,1) R(4,5)

Figure 9.24: A vector between two points

In this example, the vector led to the directed line segment fromQtoR, which we denote as

−−→

QR. We may also turn the situation around: given the two pointsQandR, we obtain the vector

h3,4_{i}because we move horizontally three units and vertically four units to get from Qto R. In
other words,−−→QR =_{h}3,4_{i}. In general, the vector−−→QRfrom the pointQ= (q1, q2)toR = (r1, r2)is

found by taking the difference of coordinates, so that

−−→

QR=_{h}r1−q1, r2−q2i.

We will use boldface letters to represent vectors, such asv =_{h}3,4_{i}, to distinguish them from
scalars. The entries of a vector are called itscomponents; in the vectorh3,4i, thexcomponent is 3
and theycomponent is 4. We use pointed bracketsh,_{i}and the termcomponentsto distinguish a
vector from a point(,)and itscoordinates. There is, however, a close connection between vectors
and points. Given a pointP, we will frequently consider the vector−−→OP from the originO toP.
For instance, ifP = (3,4), then−−→OP =_{h}3,4_{i}as in Figure9.25. In this way, we think of a pointP as
defining a vector−−→OP whose components agree with the coordinates ofP.

x y

P(3,4)

−−→

OP =h3,4i

Figure 9.25: A point defines a vector

While we often illustrate vectors in the plane since it is easier to draw pictures, different sit- uations call for the use of vectors in three or more dimensions. For instance, a vector v in n-

9.2. VECTORS 23 dimensional space,Rn, hasncomponents and may be represented as

v=_{h}v1, v2, v3, . . . , vni.

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

**Activity 9.8.**

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 thex,y, andzdirections (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, andCin the room, and complete the following.

(a) Determine the coordinates of the pointsA,B, andC. (b) Determine the components of the indicated vectors.

(i)−→OA (ii)−−→OB (iii)−−→OC (iv)−−→AB (v)−→AC (vi)−−→BC

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 vectorsu andv, we will want to be able to determine algebraically whenuandvare equal. There is an obvious set of conditions that we use.

Two vectors u = hu1, u2i andv = hv1, v2iin R2 are equal if and only if their corresponding components are equal: u1 =v1 andu2 =v2. More generally, two vectorsu =hu1, u2, . . . , uni

andv=_{h}v1, v2, . . . , vniinRnare equal if and only ifui =vifor each possible value ofi.

**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 9.9.**

Letu=_{h}2,3_{i},v=_{h−}1,4_{i}.

(a) Using the two specific vectors above, what is the natural way to define the vector sum

(b) In general, how do you think the vector suma+bof vectorsa=_{h}a1, a2iandb=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 = _{h}a1, 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_{2}v?

(e) In general, how do you think a scalar multiple of a vectora=_{h}a1, a2iinR2by a scalar
cshould be defined? how about for a scalar multiple of a vectora =ha1, a2, a3iinR3
by a scalarc? Write a formal definition of a scalar multiple of a vector based on your
intuition.

C We can now add vectors and multiply vectors by scalars, and thus we can add together scalar multiples of vectors. This allows us to definevector subtraction,v−u, as the sum ofvand−1times

u, so that

v−u=v+ (−1)u.

Using vector addition and scalar multiplication, we will often represent vectors in terms of the
special vectorsi=_{h}1,0_{i}andj=_{h}0,1_{i}. For instance, we can write the vectorha, b_{i}inR2as

ha, b_{i}=a_{h}1,0_{i}+b_{h}0,1_{i}=ai+bj,
which means that

h2,_{−}3_{i}= 2i_{−}3j.

In the context ofR3, we leti=h1,0,0i,j=h0,1,0i, andk=h0,0,1i, and we can write the vector

ha, b, c_{i}inR3as

ha, b, c_{i}=a_{h}1,0,0_{i}+b_{h}0,1,0_{i}+c_{h}0,0,1_{i}=ai+bj+ck.

The vectorsi, j, andkare called thestandard unit vectors5_{, and are important in the physical sci-}

ences.

**Properties of Vector Operations**

We know that the scalar sum1 + 2is equal to the scalar sum2 + 1. This is called thecommutative

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 twoarbitraryvectorsvandu

9.2. VECTORS 25
and add them together and see what happens. Letv=_{h}v1, v2iandu =hu1, u2i. Now we use the

fact thatv1,v2,u1, andu2are scalars, and that the addition of scalars is commutative to see that

v+u=_{h}v1, 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 follow- ing properties of vector addition and scalar multiplication.

Letv,u, andwbe vectors inRnand letaandbbe scalars. Then

1. v+u=u+v

2. (v+u) +w=v+ (u+w)

3. The vector0 = _{h}0,0, . . . ,0_{i}has the property thatv+0 = v. The vector0is called the

**zero vector**.

4. (_{−}1)v+v=0. The vector(_{−}1)v=_{−}vis called the**additive inverse**of the vectorv.
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 inR2; it is straightforward to verify that the rest of the eight properties just noted hold for all vectors inRn.

**Geometric Interpretation of Vector Operations**

Next, we explore a geometric interpretation of vector addition and scalar multiplication that al-
lows us to visualize these operations. Letu=_{h}4,6_{i}andv=_{h}3,_{−}2_{i}. Thenw=u+v=_{h}7,4_{i}, as
shown on the left in Figure9.26.

If we think of these vectors as displacements in the plane, we find a geometric way to envision vector addition. For instance, the vectoru+vwill represent the displacement obtained by follow- ing the displacementuwith the displacementv. We may picture this by placing the tail ofvat the tip ofu, as seen in the center of Figure9.26.

Of course, vector addition is commutative so we obtain the same sum if we place the tail ofu

at the tip ofv. We therefore see thatu+vappears as the diagonal of the parallelogram determined byuandv, as shown on the right of Figure9.26.

Vector subtraction has a similar interpretation. On the left in Figure9.27, we see vectorsu,v,
andw=u+v. If we rewritev=w_{−}u, we have the arrangement of Figure9.28. In other words,
to form the differencew_{−}u, we draw a vector from the tip ofuto the tip ofw.