6.3 Vector in the Plane

MagnitudeComponent form

Unit Vector

Vector is a Directed Line Segment

Terminal point

Initial point

Magnitude ( or Length): || PQ ||

Q

P

Let P = (0,0) and Q = (3,4)

To find the Magnitude || PQ ||

Direction (slope) is always important.Slope of

525169

0403 22

PQ

PQ

34

0304

PQ

Vectors equality

If two vectors equal if they have the same magnitude and direction.

P

Q

R

VRVPQ

34

PQandRVofSlope

is a vector in standard position

Vectors in Standard position have an initial point at the origin (0, 0).

PQ

Component Form of a vector

P = (p1, p2 ); Q = (q1, q2 )

Which can be labeled by just a letter.

2211 , pqpqPQ

2211 , pqpqV

Vector “V” can renamed

If || V || = 1, then V is a Unit Vector. || V || = 0 iff V is 0

2211 , pqpqV

222

1

21,

vvV

vvV

Find the Component form and Magnitude

Let URS

2,5

4,86)2(413)5(8

2

1

uu

Find the Component form and Magnitude

Let URS

2,5

4,86)2(413)5(8

2

1

uu

6,13U

Find the Component form and Magnitude

Let URS

2,5

4,86)2(413)5(8

2

1

uu

6,13U

3.14205

613 22

U

U

Vector Operations

Scalar Multiplication

Let

21, uKuKUK

30,20

65,45

56,4

KU

UK

KandU

Vector Operations

Vector Addition

Let

2211 , vuvuVU

2,6

46,24

4,2

6,4

VU

V

U

Parallelogram Law used in Addition of VectorsGraph the Vectors move the tail of one vector tothe head of the other vector.

6,4

4,2

Parallelogram Law used in Addition of VectorsGraph the Vectors move the tailof one vectorto the head of the other vector.

6,4

4,2

2,6

Properties of Vectors

U + V = V + U (Comm.)(U + V) + W = U + (V + W) (Asso.)U + 0 = U (Identity)U + (-U) = 0(Inverse)C(DU)=(CD)U (Comm.)(C + D)U = CU + DU (Dist.)1(U)=U; 0(U)=0|| cV|| =|c| x ||V||

How to Find the Unit Vector

Let

657,

6547,4

651

74

7,4||||

7,4

22

vv

v

Standard Unit Vector

Writing the Unit Vector as Standard Unit Vector.

i =j =

0,1

1,0

657,

654

||||

7,4

vvu

v

0,0 1

1

i

j

jiu657

654

Direction Angle of a Unit Vector

What is the coordinate of the intersection of the vector and

unit circle?

Direction Angle of a Unit Vector

What is the slope of the vector?What functionIs rise over run?

sin,cos

ab

bjai

Direction Angle of a Unit Vector

What is the slope of the vector?What functionIs rise over run?

sin,cos

ab

bjai

ab

cossintan

ji sincos

Direction of a Vector can be found if it is not a Unit Vector

jvivvv sincossin,cos

ab

vv

cossin

tan

Homework

Page 436 – 437# 1, 7, 15, 19,

25, 31, 37, 43, 49, 55, 61, 67, 73

Homework

Page 436 – 437# 32, 38, 54, 62,

70, 80