6.3 Vector in the Plane

Magnitude

Component 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

3

4

03

04

PQ

Vectors equality

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

P

Q

R

VRVPQ

3

4PQandRVofSlope

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(4

13)5(8

2

1

u

u

Find the Component form and Magnitude

Let URS

2,5

4,86)2(4

13)5(8

2

1

u

u

6,13U

Find the Component form and Magnitude

Let URS

2,5

4,86)2(4

13)5(8

2

1

u

u

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 to

the head of

the other

vector.

6,4

4,2

Parallelogram Law used in Addition of VectorsGraph the

Vectors

move the tail

of one vector

to 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

65

7,

65

47,4

65

1

74

7,4

||||

7,4

22

v

v

v

Standard Unit Vector

Writing the Unit Vector as Standard Unit Vector.

i =

j =

0,1

1,0

65

7,

65

4

||||

7,4

v

vu

v

0,0 1

1

i

j

jiu65

7

65

4

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 function

Is rise over run?

sin,cos

a

b

bjai

Direction Angle of a Unit Vector

What is the slope of the vector?

What function

Is rise over run?

sin,cos

a

b

bjai

a

b

cos

sintan

ji sincos

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

jvivvv sincossin,cos

a

b

v

v

cos

sintan

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