Vectors in Plane

Objectives of this Section

• Graph Vectors

• Find a Position Vector

• Add and Subtract Vectors

• Find a Scalar Product and Magnitude of a Vector

• Find a Unit Vector

• Find a Vector from its Direction and Magnitude

• Work With Objects in Static Equilibrium

A vector is a quantity that has both magnitude and direction.

Vectors in the plane can be represented by arrows.

The length of the arrow represents the magnitude of the vector.

The arrowhead indicates the direction of the vector.

P

Q

Initial Point

Terminal Point

Directed line segmentPQ

.point the topoint for the distance theis

segment line directed theof magnitude The

QP

PQ

. to from is ofdirection The QPPQ

If a vector v has the same magnitude and the same direction as the directed line segment PQ, then we write

PQv

The vector v whose magnitude is 0 is called the zero vector, 0.

v w if they have the same magnitude and direction.

Two vectors v and w are equal, written

vw

v w

v

wv w

Initial point of v

Terminal point of w

Vector Addition

Vector addition is commutative.

v w w v Vector addition is associative.

u v w u v w

v 0 0 v v

v v 0

Multiplying Vectors by Numbers

If is a scalar (a real number) and is a vector,

the is defined as

v

scalar product v1.

.

If > 0, the product is the vector

whose magnitude is times the magnitude

of and whose direction is the same as

v

v v

2.

.

If < 0, the product is the vector

whose magnitude is times the magnitude

of and whose direction is opposite that of

v

v v

3. . If = 0 or if , then v 0 v 0

Properties of Scalar Products

0 1 1v 0 v v v v

v v v v w v w

v v

Use the vectors illustrated below to graph each expression.

v

w

u

v w

v w

wv - and 2

v

2v

w

w

2v w

2v

w

If is a vector, we use the symbol to

represent the of

v v

magnitude v.

vv

vv

0vv

v

v

(d)

(c)

ifonly and if 0 b)(

0 (a)

thenscalar, a is if and vector a is If

A vector for which is called a

.

u u

unit vector

1

Let i denote a unit vector whose direction is along the positive x-axis; let j denote a unit vector whose direction is along the positive y-axis. If v is a vector with initial point at the origin O and terminal point at P = (a, b), then

v i j a b

ai

bj

a

P = (a, b)

v = ai

+ bjb

The scalars a and b are called components of the vector v = ai + bj.

ectorposition v the toequal is

then, If .,point terminal

and origin, y thenecessarilnot ,,

point initialth vector wia is that Suppose

21222

111

v

v

v

PPyxP

yxP

v i j x x y y2 1 2 1

.4,3 and 1,2 if

vector theofector position v theFind

2121

PPPPv

v i j x x y y2 1 2 1

v i j 3 2 4 1( )

v i j 5 3

P1 2 1 ,

P2 3 4 ,

5 3,

O

v = 5i + 3j

Equality of Vectors

Two vectors v and w are equal if and only if their corresponding components are equal. That is,

If and +

then if and only if and

v i j w i j

v w

a b a b

a a b b1 1 2 2

1 2 1 2

,

.

Let and be two

vectors, and let be a scalar. Then,

v i j w i j a b a b1 1 2 2

v w i j

v w i j

v i j

v

a a b b

a a b b

a b

a b

1 2 1 2

1 2 1 2

1 1

12

12

If and , find

(a) (b)

v i j w i j

v w v w

3 2 4

(a) v w i j i j 3 2 4

3 4 2 1i j

i j3 (b) v w i j i j 3 2 4

3 4 2 1i j 7i j

If and , find

(a) 2 (b)

v i j w i j

v w v

3 2 4

3

(a) 2 3 2 3 2 3 4v w i j i j

6 4 12 3i j i j

6 7i j

(b) 2v i j 3 3 22 2

13

Unit Vector in Direction of v

For any nonzero vector v, the vector

uvv

is a unit vector that has the same direction as v.

Find a unit vector in the same direction as v = 3i - 5j.

v 3 52 2( ) 9 25 34

uvv

i j 3 534

334

534

i j

3 3434

5 3434

i j