homework, page 519

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Homework, Page 519

1.

Find the dot product of and .u v

5,3 , 12,4u v

5 12 3 4 60 12 72u v

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Homework, Page 519

5.

Find the dot product of and .u v

4 9 , 3 2u i j v i j

4, 9 , 3, 2u v

4 3 9 2 12 18 30u v

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Homework, Page 519

9.

Use the dot product to find .u

5, 12u

2 2

cos 1u u u u u u

5 5 12 12 25 144 169u u

169 13u

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Homework, Page 519

13.

Find the angle between the vectors.4, 3 , 1,5u v

1cos cos cosu v u v

u v u vu v u v

4 1 3 5 4 15 11u v

2 24 3 16 9 25 5u

2 21 5 1 25 26v

1 1 11cos cos 115.560

5 26

u v

u v

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Homework, Page 519

17.

Find the angle between the vectors.3 3 , 2 2 3u i j v i j

1cos cos cosu v u v

u v u vu v u v

3 2 3 2 3 6 6 3u v

2 23 3 9 9 18 3 2u

222 2 3 4 12 16 4v

1 1 6 6 3cos cos 165

3 2 4

u v

u v

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Homework, Page 519

21.

Find the angle between the vectors.

8 3 5 4 24 20 4u v

u

(8, 5)

v

(-3, 4)

2 28 5 64 25 89u

2 23 4 9 16 25 5v

1 1 4cos cos 94.865

89 5

u v

u v

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Homework, Page 519

25.v

Find the vector projection of of u onto v. Then write u as a sum

of two orthogonal vectors, one of which is proj u .

2 28,3 , 6, 2 6 2 36 4 40u v v

8,3 , 6, 2u v

2 2

8 6 3 2 48 66, 2 6, 2

4040v

u vproj u v

v

42 21 21 6, 2 6, 2 3, 1

40 20 10

63 21,

10 10vproj u

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Homework, Page 519

25.v

Find the vector projection of of u onto v. Then write u as a sum

of two orthogonal vectors, one of which is proj u .

8,3 , 6, 2u v

2 21, 3 1 3 10w w ����������������������������

2w

u wproj u w

w

��������������

������������� �������������� �

��������������

8,3 1, 3 8 9 17u w ������������� �

217 17 17 51

1, 3 1, 3 ,10 10 1010

wproj u

��������������

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Homework, Page 519

25.v

Find the vector projection of of u onto v. Then write u as a sum

of two orthogonal vectors, one of which is proj u .

8,3 , 6, 2u v

63 21 17 51, ,

10 10 10 10v wproj u proj u ������������� �

80 30, 8,3

10 10

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Homework, Page 519Find the interior angles of the triangle with the given vertices.

29. 4,5 , 1,10 , 3,1 1 4 ,10 5 5,5 3 4 ,1 5 7, 4u v

2 21 3,10 1 2,9 5 5 50 5 2w u

����������������������������

2 2 2 27 4 65 2 9 85v w

������������� �

1 11

35 20cos cos 74.745

5 2 65

u v

u v

1 12

10 45cos cos 57.529

5 2 85

u w

u w

������������� �������������� �

3 180 47.426

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Homework, Page 519Determine whether the vectors are parallel, orthogonal, or neither.

33. 10 35,3 , ,

4 2u v

10 3 25 95 3 17

4 2 2 2u v

2 25 3 25 9 34u

2 210 3 25 9 34

4 2 4 4 2v

1 1 17cos cos 180 Parallel

3434

2

u v

u v

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Homework, Page 519Determine whether the vectors are parallel, orthogonal, or neither.

37. 3,4 , 20,15u v

3 20 4 15 60 60 0 Orthogonalu v

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Homework, Page 519Find (a) the x- and y-intercepts (A and B) of the line and

(b) the coordinates of the point P so that the unit vector AP is perpendicular to the line.

41. 3 7 21x y 3 7 21 3 0 7 21 3 3 7 0 21 7x y y y x x

2 23,7 3 7 9 49 58u u

3 58 7 58,

58 58

u

u

3 58 7 587 ,0 6.606,0.919

58 58P

7 0 , 0 3 7,3 , 0AB u a b AB u ��������������������������������������������������������

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Homework, Page 51945. Ojemba is sitting on a sled on the side of a hill inclined 60º. The combined weight of Ojemba and the sled is 160 lb. What is the magnitude of the force required for Mandisa to keep the sled from sliding down the hill?

sin 60F wt 160sin 60 80 3 138.564 lbF

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Homework, Page 51949. Find the work done lifting a 2600-lb car 5.5 feet.

W F d����������������������������

2600 lb 5.5 ft 14,300 ft-lbW

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Homework, Page 51953. Find the work done by force F of 30 lb acting in the direction (2, 2) in moving an object 3 ft from (0, 0) to a point in the first quadrant along the line y = ½ x.

11 12,2 45 tan 26.565

2 2y x

cos 30 3cos 45 25.565 85.381 ft-lbW F d

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Homework, Page 51957. Use the component form of vectors to prove the following.

Let , , , , , and .u a b v d e w f g c c ������������� �

0 0 0 0 0 0 0 0a u u a b u

,b u v w u v u w v w d f e g ������������������������������������������������������������������������������������������� �������

u v w ad af be bg ������������� � u v u w ad be af bg

������������� �

ad af be bg ad be af bg u v w u v u w ����������������������������������������������������������������� �����

,c u v w u w v w u v a d b e ������������������������������������������������������������������������������������������� �������

u v w af df bg eg ������������� �

u w v w af bg df eg ��������������������������������������� ���

af df bg eg af bg df eg u v w u w v w ����������������������������������������������������������������� �����

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Homework, Page 51957. Continued

,cv cd ce

,cu ca cb d cu v u cv c u v

, ,cu v ca cb d e cad cbe

, ,u cv a b cd ce acd bce

, ,u v a b d e ad be c u v c ad be cad cbe

cad cbe acd bce cad cbe cu v u cv c u v

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Homework, Page 51961. If 0, then and are perpendicular. Justify your answer.u v u v

1

True. If 0 and 0 and 0, the cos 0

cos 0 90 and .

u v u v

u v

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Homework, Page 51965.

a.

b.

c.

d.

e.

v

3 3Let , and 2,0 . Which of the following is

2 2

equal to proj ?

u v

u

3 ,02

3,0

3 ,02

3 3,2 2

3 3,2 2

2

2

proj 2

3 0 3proj 2,0 ,022

v

v

u vu v v

v

u

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

6.3

Parametric 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 line

through 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23

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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24

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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 25

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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 26

Example Graphing Parametric Equations

2

For the given parametric interval, graph the parametric equations

2, 3 .

(a) 3 1 (b) 2 3 (c) 3 3

x t y t

t t t

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27

Example Graphing Parametric Equations

2

For the given parametric interval, graph the parametric equations

2, 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 parametric

curve 1, 2 , .x t y t t

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Example Eliminating the Parameter

Eliminate the parameter and identify the graph of the parametric

curve 3cos , 3sin , 0 2 .x t y t t

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 30

Example Finding Parametric Equations for a Line

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

and ( 3,6).

A

B

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 31

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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 32

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?

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

Ball does not clear the fence.

Slide 6- 33

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 34

Homework

Homework Assignment #5Review Section 6.3Page 530, Exercises: 1 – 65 (EOO)

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

6.4

Polar Coordinates

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 36

Quick Review Solutions

1. Determine the quadrants containing the terminal side of the angle:

4 / 3

2. Find a positive and negative angle coterminal with the given angle:

/ 3

3. Write a s

II

5 /3,

tandard form e

7 /3

qu

2 2

ation for the circle with center at ( 6,0)

and a radius ( 6) 1f 6o 4. x y

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 37

Quick Review Solutions

Use the Law of Cosines to find the measure of the third side of the given triangle.

4.40º

8 10

5.

35º

6 11

6.47

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 38

What you’ll learn about

Polar Coordinate System Coordinate Conversion Equation Conversion Finding Distance Using Polar Coordinates

… and whyUse of polar coordinates sometimes simplifies complicated rectangular equations and they are useful in calculus.

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The Polar Coordinate System

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Example Plotting Points in the Polar Coordinate System

Plot the points with the given polar coordinates.

( 1, (a) (2 (3, 45 )3 / 4 (b) (c, / ) ) 3)P RQ

x

y

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 41

Finding all Polar Coordinates of a Point

Let the point have polar coordinates ( , ). Any other polar

coordinate of must be of the form ( , 2 ) or

( , (2 1) ) where is any integer. In particular, the

pole has polar coordinates (0

P r

P r n

r n n

, ), where is any angle.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 42

Coordinate Conversion Equations

2 2 2

Let the point have polar coordinates ( , ) and rectangular

coordinates ( , ). Then

cos , sin , , tan .

P r

x y

yx r y r x y r

x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 43

Example Converting from Polar to Rectangular Coordinates

Find the rectangular coordinate of the point with the polar

coordinate (2, 7 / 6).

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 44

Example Converting from Rectangular to Polar Coordinates

Find two polar coordinate pairs for the point with the rectangular

coordinate (1, 1).

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Example Converting from Polar Form to Rectangular Form

Convert 2sec to rectangular form.r

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 46

Example Converting from Polar Form to Rectangular Form

2 2

Convert 2 3 13 to polar form.x y