10.2

Vectors in the Plane

Quick ReviewLet (1,3) and (4,7).

1. Find the distance between the points P and Q.

2. Find the slope of the line segment PQ.

3. If (3, ), determine so that segments and are colinear.

4. If (3, ), deter

P Q

R b b PQ RQ

R b

mine so that segments and are perpendicular.

5. Determine the missing coordinate so that the four points form a

parallelogram . (1,1), (3,5), (8, ), (6, 2)

b PQ RQ

ABCD A B C b D

5

3

4

3

17b

4

31b

6b

Quick Review6. Find the velocity and acceleration of a particle moving along a line if

its position at time is given by ( ) cos .

7. A particle moves along the -axis with velocity ( ) 2 6 for 0.

If its pos

t s t t t

x v t t t

ition is 10 when 0, where is the particle when 6?

8. Find the length of the curve defined parametrically by sin 2

and cos3 for 0 2 .

x t t

x t

y t t

ttttv cossin ttta sin2cos

826 s

28937.15L

What you’ll learn about Two-Dimensional Vectors Vector Operations Modeling Planar Motion Velocity, Acceleration, and Speed Displacement and Distance Traveled

Essential QuestionHow does the mathematics of vectors help usjump from one to two dimensions (and eventuallyhigher)?

Two-Dimensional Vectors A is an ordered pair of real numbers, denoted in

as , . The numbers and are the of the

vector . The of the vector ,

a b a b

a b

two - dimensional vector

component form components

standard representation

v

v is the arrow from

the origin to the point , . The of , denoted , is the length

of the arrow, and the is the direction in which the arrow is

pointing. The vector 0,0 , call

a b

magnitude

direction of

0

v v

v

ed the , has zero length and

no direction.

zero vector

Magnitude of a Vector

2 2

The or of the vector , is the nonnegative

real number , .

a b

a b a b

magnitude absolute value

Direction Angle of a Vector The of a nonzero vector is the smallest nonnegative

angle formed with the positive -axis as the initial ray and the

standard representation of as the terminal ray.

xdirection angle v

v

Head Minus Tail (HMT) Rule

1 1 2 2

2 1 2 1

If an arrow has initial point ( , ) and terminal point ( , ),

it represents the vector , .

x y x y

x x y y

Example Finding Magnitude and Direction1. Find the magnitude and the direction angle if the vector v = .3 ,1

22 is of Magnitude vv 1 3

231

3 ,1

1

3

tan 3

1

2

603

Example Finding Magnitude and Direction2. Find the component form of the vector with the given magnitude

that forms the given directional angle with the positive x-axis.100 ,5

.9244 ,868.0

868.0

924.4100

80sin x

5

5

80sin5x 924.4

80

80cos x

5 80cos5x 868.0

924.4 ,868.0

Vector Addition and Scalar Multiplication

1 2 1 2

1 1 2 2

Let = , and = , be vectors and let be a real number (scalar).

The sum (or resultant) of the vectors and is the vector

, .

The and the vector is

u u v v k

u v u v

k k u

product of the scalar

u v

u v

u v

k u

u1 2 1 2, , .

The is ( 1) . We define vector subtraction by

+( ).

u ku ku

opposite of a vector v v v

u v u v

Performing Operations on Vectorsfollowing. theFind .8 ,4 and 2 ,1Let 2. vu

vu 3 2 a. 8 ,432 ,12

4 ,2 42 ,12 ,10 28vu b.

8 ,42 ,1 2 ,1 8 ,4 ,5 6

v2

1 c. 8 ,4

2

1 4 ,2

22 2 4 164 20 52

Properties of Vector Operations Let , , be vectors and , be scalars.

1. + = +

2. ( + ) + = + ( + )

3. + =

4. + ( ) =

5. 0 =

6. 1

7. ( ) ( )

8. ( + )

9. ( )

a b

a b ab

a a a

a b a b

u v w

u v v u

u v w u v w

u 0 u

u u 0

u 0

u u

u u

u v u v

u u u

Example Finding Ground Speed and DirectionA plane flying due east at 400 mph in still air, encounters a 50-mph

tail wind acting in the direction 60 north of east. The plane holds

its compass heading due east but, because of the wind, acquire

s

a new ground speed and direction. What are they?

3.

60

50

400

air stillin plane of velocity Let u 400u wind tailof velocity Let v 50v

0 ,400u 60sin50 ,60cos50v

352 ,25vu ,425 325

vu 22 425 325427.2 mph

425

325

tan 1 325

4255.8 east ofnorth

Velocity, Speed, Acceleration, and Direction of Motion

Suppose a particle moves along a smooth curve in the plane so that its position

at any time is ( ( ), ( )), where and are differentiable functions of t.

1. The particle's is ( )

t x t y t x y

tposition vector r

2

2

( ), ( ) .

2. The particle's is ( ) , .

3. The particle's is the magnitude of , denoted | |. Speed is a scalar.

4. The particle's is ( ) ,

x t y t

dx dyt

dt dt

d xt

dt

velocity vector

speed

acceleration vector

v

v v

a2

2 .

5. The particle's is the .

d y

dt

direction of motion direction vectorv

v

Displacement and Distance TraveledSuppose a particle moves along a path in the plane so that its velocity

at any time t is , , 21 tvtvtv where v1 and v2 are integrable

functions of t.

The displacement from t = a to t = b is given by the vector

. ,

21 b

a

b

adttvdttv

The preceding vector is added to the position at t = a to get the

position at t = b. The distance traveled from t = a to t = b is

b

adttv

b

adttvtv

22

21

Example Finding Displacement and Distance Traveled

4. A particle moves in the plane with velocity vector

. sin2 , cos 3 tttttv At t = 0, the particle is at the point (1, 5).

a. Find the position of the particle at t = 4.

dtdt , Disp.4

0

4

0 tt cos 3 tt sin 2

61 ,8 5 ,1at is particle the,0at t

61 ,85 ,1 11 ,9

Example Finding Displacement and Distance Traveled

4. A particle moves in the plane with velocity vector

. sin2 , cos 3 tttttv At t = 0, the particle is at the point (1, 5).

b. What is the total distance traveled by the particle from t = 0 to t = 4?

4

0

22 Dist. dttt cos 3 tt sin 2

533.33

Pg. 545, 10.2 #1-47 odd