homework, page 511

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Homework, Page 511

1.

Prove that and are equivalent by showing that they represent the same vector.RS PQ

4,7 , 1,5 , 0,0 , and 3,2R S P Q

4,7 , 1,5 , 0,0 , and 3, 2

1 4 , 5 7 3, 2

3 0 , 2 0 3, 2

R S P Q

RS

PQ RS PQ

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Homework, Page 511

5.

Let 2,2 , 3,4 , 2,5 , and 2, 1 . Find the component form

and magnitude of the vector.

P Q R S

PQ

2 2

3 2 , 4 2 5,2 5,2

5 2 25 4 29 29

PQ PQ

PQ PQ

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Homework, Page 511

9.

Let 2,2 , 3,4 , 2,5 , and 2, 1 . Find the component form

and magnitude of the vector.

P Q R S

2QS

2 2

2 2 2 3 , 8 4 2 1, 12 2, 24 2 2, 24

2 2 24 4 576 580 2 145 2 2 145

QS QS

QS QS

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Homework, Page 511

13.

Let 1,3 , 2,4 , and 2, 5 . Find the component form

of the vector.

u v S

u v

1 2 , 3 4 1,7 1,7u v u v

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Homework, Page 511

17.

Let 1,3 , 2,4 , and 2, 5 . Find the component form

of the vector.

u v S

2 3u w

2 3 2 1,3 3 2, 5

2,6 6, 15

2 6 , 6 15

4, 9

2 3 4, 9

u w

u w

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Homework, Page 511Find a unit vector in the direction of the given vector.

21. 2,4u

2 22,4 2 4 4 16 20 2 5

2,4 1 2 5 2 5, ,5 52 5 5 5

u u

uu

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Homework, Page 511Find a unit vector in the direction of the given vector. Write the answer in (a) component form and (b) as a linear combination of the standard unit vectors.

25. 2,1u

2 22,1 2 1 4 1 5

2,1 2 1 2 5 5, ,5 55 5 5

2 5 5 2 5 5, 5 5 5 5

u u

uu

u ua b i ju u

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Homework, Page 511Find the component form of the vector.

29.

18 cos 25 sin 25 16.314,7.607u

x

y

25

18

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Homework, Page 511Find the magnitude and direction angle of the vector.

33. 3,4

2 2

1

3,4 3 4 9 16 25 5

33,4 cos ,sin 3 5cos cos5

cos 0.6 53.130 5; 53.130

r

r

r

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Homework, Page 511Find the magnitude and direction angle of the vector.

37. 7 cos135 sin135i j

7 cos135 sin135 7; 135i j r

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Homework, Page 51141. An airplane is flying on a bearing of 335º at 530 mph. Find the component form of the velocity of the airplane.

360 335 25 25 90 115530 cos115 sin115 223.987,480.343

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Homework, Page 51145. A basketball is shot at a 70º angle with the horizontal direction with initial speed 10 fps. (a) Find the component form of the initial velocity.

(b) Give an interpretation of the horizontal and vertical components of the velocity.The horizontal component is the horizontal speed, in the absence of air resistance, the ball will maintain. The vertical component is the initial vertical velocity of the ball, which will change continuously due to the force exerted by gravity on the ball.

10 cos70 sin 70 3.420,9.397

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Homework, Page 511Find the direction and magnitude of the resultant force.

49. A force of 50 lb acts on an object at an angle of 45º. A second force of 75 lb acts on the object at an angle of –30º.

2 2

1

50 cos 45 ,sin 45 75 cos 30 ,sin 30

35.355,35.355 64.952, 37.5 100.307, 2.145

100.307 2.145 100.330

2.145tan 1.225 =100.330 lb at 1.225100.330

r

r

R

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Homework, Page 51153. A ship heads due south with the current flowing northwest. Two hours later the ship is 20 miles in the direction 30º west of south from the original starting point. Find the speed with no current of the ship and the rate of the current.

2 cos270 ,sin 270 0,2

2 cos135 ,sin135 1.414 ,1.414

20 cos240 ,sin 240 10, 17.320

1.414 102 1.414 17.3202 27.320

13.660 7.071

s s s

c c c c

r

cs cs

s kts c kts

180

315

210

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Homework, Page 51157. Which of the following is the magnitude of the vector (2, –1)?A. 1B.C.D.E. 5

35 55

2 22 1 4 1 5

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6.2Dot Product of Vectors

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What you’ll learn about

The Dot Product Angle Between Vectors Projecting One Vector onto Another Work

… and whyVectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

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Dot Product

1 2

1 2 1 1 2 2

The or of , and

, is cos .

u u

v v u v u v u v

dot product inner product

u

v u v

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Properties of the Dot Product

Let u, v, and w be vectors and let c be a scalar.1. u·v=v·u2. u·u=|u|2

3. 0·u=04. u·(v+w)=u·v+u·w (u+v) ·w=u·w+v·w5. (cu) ·v=u·(cv)=c(u·v)

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Example Finding the Dot Product

Find the dot product: 4,3 1, 2

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Angle Between Two Vectors

1

If is the angle between the nonzero vectors and , then

cos and cos and cosu v

u v

u v u vu vu v u v

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Example Finding the Angle Between Vectors

Find the angle between the vectors 3,2 and 1,0 .

u v

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Example Finding a Force to Overcome Gravitational Pull

Suppose Rafaela is sitting on a sled on a 45º slope. If she and the sled have a combined weight of 140 lb, how much force must Juan apply to a rope tied to the sled to prevent its sliding down the hill?

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Orthogonal Vectors

cos90 0 0u v u v

u v

Perpendicular vectors are sometimes referred to as orthogonal vectors. The vectors u and v are orthogonal if and only if u·v = 0.

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Projection of u and v

2

2 2

If and are nonzero vectors, the projection of onto

is proj .

cos

cosproj cos

u v

u vu

v

v

v

u v u

u vv u vv

u v

u v vu v vv v

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example, Page 520Find the projection of u onto v. Then write u as a sum of two orthogonal vectors, one of which is projv u.

26. u = (3, –7), v = (–2, –6)

Slide 6- 26

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Work

If is a constant force whose direction is the same as

the direction of AB, then the done by in

moving an object from to is | || AB | . If the force is applied in a directi

A B

work W

W

F

F

Fon other than the direction

of motion, then the work done is

cosW F AB F AB

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Example Finding the Work Done by a Constant Force

Find the work done by a force F of 50 lb acting in the direction (2, 3) in moving an object five feet from (0, 0) to a point in the first quadrant along the line y = x.

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Homework

Homework Assignment #18 Read Section 6.3 Page 519, Exercises: 1 – 65 (EOO) Quiz next time

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6.3Parametric Equations and Motion

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Quick Review

1. Find the component form of the vectors

(a) OA, (b) OB, and (c) AB where O is the origin,(3,2) and (-4, -6).

2. Write an equation in point-slope form for the linethrough the points (3,2) and (-4,-6

A B

2

).3. Find the two functions defined implicitly by 2 .4. Find the equation for the circle with the center at (2,3)and a radius of 3.5. A wheel with radius 12 in spins at the rate 400 rpm. Find the a

y x

ngular velocity in radians per second.

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Quick Review Solutions

1. Find the component form of the vectors

(a) OA, (b) OB, and (c) AB where O is the origin,(3,2) and ( 4, 6).

2. Write an equation in point-sl

(a) 3

ope

,

form for the line

thro

2 (b) 4, 6 (c) 7,

ugh

8A B

2

2 2

82 ( 3)7

2 ;

the points (3,2) and ( 4, 6).

3. Find the two functions defined implicitly by 2 .

4. Find the equation for the circle with the center at (2,3)

and a radius of 3.

2

2 3 9

5. A

y x

y x

y x

y x

x y

wheel with radius 12 in spins at the rate 400 rpm. Find the angular velocity in radians per second. 40 / 3 rad/sec

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What you’ll learn about

Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher

… and whyThese topics can be used to model the path of an object such as a baseball or golf ball.

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Parametric Curve, Parametric Equations

The graph of the ordered pairs (x,y), where x = f(t) and y = g(t) are functions defined on an interval I of t-values, is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval.

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Example Graphing Parametric Equations

2

For the given parametric interval, graph the parametric equations2, 3 .

(a) 3 1 (b) 2 3 (c) 3 3 x t y t

t t t

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Example Graphing Parametric Equations

2

For the given parametric interval, graph the parametric equations2, 3 .

(a) 3 1 (b) 2 3 (c) 3 3 x t y t

t t t

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Example Eliminating the Parameter

Eliminate the parameter and identify the graph of the parametriccurve 1, 2 , .x t y t t

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Example Eliminating the Parameter

Eliminate the parameter and identify the graph of the parametriccurve 3cos , 3sin , 0 2 .x t y t t

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Example Finding Parametric Equations for a Line

Find a parametrization of the line through the points (2,3)and ( 3,6).

AB

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Example Simulating Horizontal Motion

3 2

Gary walks along a horizontal beam with the coordinate of

his motion given by 0.1 20 110 85 where

0 12. Estimate the times when Gary changes dierection.

x t t t

t

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Example Simulating Projectile MotionMatt hits a baseball that is 3 ft off the ground at an angle of 30° above the horizontal with an initial velocity of 125 fps. Does the ball clear a 20 ft fence 400 ft from the plate?