Vectors in the Plane

Peter Paliwoda

Introduction to Vectors• Quantities such as force and velocity involve

both magnitude and direction• Such quantities cannot be characterized by

single real numbers• To represent such a quantity, you can use a

directed line segment shown in Fig. 6.15

Introduction to Vectors

• The directed line segment PQ has initial point P and terminal point Q. Its magnitude or (length) is denoted by found by Distance Formula QP

Introduction to Vectors• Two directed line segments that have same

magnitude and direction are equivalent• For example, segments shown below in Figure

6.16 are all equivalent• Vectors are denoted by lowercase, boldface

letters such as u, v, and w. Ex. v= PQ

Introduction to Vectors• Let u represent line segment from P=(0,0) to Q=(3,2)• Let v represent line segment from R=(1,2) to S=(4,4)• Show that u=v

130203 22 QP

132414 22 SR

• Moreover, both lines have the same direction because they are both directed toward upper right on lines having slope of 2/3. Therefore PQ and RS have the same magnitude and direction, thus u=v

Component Form of a Vector

• A vector whose initial point is the origin (0,0) can be uniquely represented by the coordinates of its terminal point (v1, v2)

• This is the component form of a vector v, written as v= v1, v2

Finding the Component Form of a Vector

• Initial point (4,-7) and terminal point (-1,5)• v1=-1-4=-5; v2=5-(-7)=12, so v= -5,12• So v= -5,12 and the magnitude of v is

13169125 22 v

Vector Operations

• Two basic operations are scalar multiplication and vector addition

Vector Operations• Let v= -2,5 and w= 3,4 . Find 2v, w-v, v+2w

13,4

85,62

8,65,2

42,325,2

4,325,22

1,554,23

.10,452,225,222

wv

vw

v

Unit Vectors• In many applications its useful to find a unit

vector that has the same direction as a given nonzero vector v. v

vv

vvectorunitu

1

• Example, find the unit vector of v= -2,5

129

29

29

25

29

4

29

5

29

2

29

5,

29

2

29

5,2

52

5,2

22

22

magnitudeCheck

v

v

Example Problem