Vectors in the PlaneObjectives:•Define a vector•What are the basic terms associated with vectors?

DefinitionA vector is a quantity that has magnitude (size) and direction. It is used to measure quantities, such as force, velocity and acceleration.

A vector is a displacement

A vector is a translation

Traveling northeast at a speed of 60 mph.

60 mph NE

Example

ExampleTraveling west at a speed of 30 mph.

30 mphwes

t

A vector is represented using a directed line segment.

v

DefinitionVectors that point in the opposite direction and have the same length are opposite vectors.

u vu v

2 2a b

DefinitionThe magnitude of a vector is represented by its length. If , then the magnitude is :

,,

a bu a b T

2 2 u a b

ab

DefinitionVectors may be multiplied by a scaler. A scaler is a real number. It has magnitude (size) only.u v

2 w v

12

u vw

Given rv

1r

is a scalarmultiple

ofv

Describe the effect on the vector when:

.

0 1r

0r

“stretches” the vector “shrinks” the vector “reflects” the vector through a point

DefinitionVectors are parallel if and only if they are scalar Multiples of each other.

Given Points A(3,2) B(6,4)C(4, -2) and D(2,-1)

v AB

and w CD

Express each as a translation1)v 2)w

3)2v 4) 3w5)v w 6)v w

Do Now:

Every vector consists of at least 2 component vectors:1 in the x direction and 1 in the y direction This vector is in standard position.Component Vectors

Aim: How do we represent vectors algebraically ?

Important IdeaA vector may be represented by subtraction of the ordered pairs.

u(2,3)(4,5)

(2,2)u2,2

ExampleFind the component form of the vector v and sketch the vector with its initial point at the origin.

(2,3)

(5,5)

(3,2)

3,2

DefinitionVector Addition:

,u v a c b d

,u a b

,v c dIf and then

u v is also

calledor resultant vector.

the vector sum

u

vu v

Try This2,8u

5, 3v If and ,

find u v

7,5

DefinitionVector Subtraction: If

,u a b and , then: ,v c d

,u v a c b d

Vector Subtraction:,u v a c b d

DefinitionScaler Multiplication:

,u a bIf and r is a real

,ru ra rb

number, then:

Try This2, 3u

If and r=2, find ru

4, 6

and sketch its graph in standard position.

Definition1,0i and 0,1j

are unit vectors. Any vector can be written as a scaler times i plus a scaler times j.

Example2,3 2,0 0,3

2 1,0 3 0,1

2 3i j

Try ThisWrite using unit vectors.

5, 3

5i-3j

10cos130 ,10sin130

6.43 , 7.66

A Force F of 10 Newtons acts at an angle of with the positive x-axis. Express F as a vector in component form.

o130

Given A(0,4) and B(6,1). Find the coordinates of point P that is 2/3 of the way from A to B.

NEXT DO NOW

Try This

Solution:OP OA AP

20,43

AB

20,4 6, 33

0,4 4, 2

4,2

Given A(8,4) and B(-1,7). Find the coordinates of point P that is 2/3 of the way from A to B.

Another One

O28,4328,4 9,33

8,4 6,2

2,6

P

OA AP

AB

Describing Position:, 0,3 2, 1 x y t

The position of an object at time t is given as,x y and 0,3 are standard

position vectorsWhere is the object at t=0? t=1 t=2? t = 3? t = 4?

t R(t)01234

, 0,3 2, 1 x y t

0,32,24,16,08, 1

, 0,3 2, 1 x y t

What is the speed of the Object?

What direction is the Object moving?

0 0 1 2, , ,x y x y t v v

Vector Equation of a lineGiven standard position vectors

and The equation of a line is:

,x y0 0,x y

0 0,x y

1 2,v vis the Initial Position is the Velocity or Direction

Vector 1 2,v v is the Speed

, 4,9 1, 2 x y tFlyWhat is the speed of the Fly?

What direction is the Fly movingHow are these Flies different?

, 4,9 2, 4 x y t, 4,9 1,2 x y t

Fly 2Fly 3

Can this spider catch any of the Flies? Explain.

, 1,1 1,1 x y t

An object moves with constant velocity. At t = 0 it is at A(5,3) and at t = 3 it is at B(−4,15). a) find its velocity b) find its speed c) Write a vector equation describing the objects movement.

Example