homework, page 519
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Homework, Page 519. 1. Homework, Page 519. 5. Homework, Page 519. 9. Homework, Page 519. 13. Homework, Page 519. 17. Homework, Page 519. 21. Homework, Page 519. 25. Homework, Page 519. 25. Homework, Page 519. 25. Homework, Page 519. - PowerPoint PPT PresentationTRANSCRIPT
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 2
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 3
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8
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
��������������
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12
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 ��������������������������������������������������������
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 15
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 16
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17
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 ����������������������������������������������������������������� �����
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 19
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 20
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 28
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the parametric
curve 1, 2 , .x t y t t
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 29
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 39
The Polar Coordinate System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 40
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).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 45
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