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C H A P T E R 1 0Analytic Geometry in Three Dimensions
Section 10.1 The Three-Dimensional Coordinate System . . . . . . . . 888
Section 10.2 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . 897
Section 10.3 The Cross Product of Two Vectors . . . . . . . . . . . . . 905
Section 10.4 Lines and Planes in Space . . . . . . . . . . . . . . . . . 912
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927
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C H A P T E R 1 0Analytic Geometry in Three Dimensions
Section 10.1 The Three-Dimensional Coordinate System
Vocabulary Check
1. three-dimensional 2. xy-plane, xz-plane, yz-plane
3. octants 4. Distance Formula
5. 6. sphere
7. surface, space 8. trace
�x1 � x2
2,
y1 � y2
2,
z1 � z2
2 �
■ You should be able to plot points in the three-dimensional coordinate system.
■ The distance between the points and is
.
■ The midpoint of the line segment joining the points and is
■ The equation of the sphere with center and radius is
■ You should be able to find the trace of a surface in space.
�x � h�2 � � y � k�2 � �z � j�2 � r2.
r�h, k, j�
�x1 � x2
2,
y1 � y2
2,
z1 � z2
2 �.
�x2, y2, z2��x1, y1, z1�d � ��x2 � x1�2 � � y2 � y1�2 � �z2 � z1�2
�x2, y2, z2��x1, y1, z1�
2. C��2, 3, 0�A�6, 2, �3�, B�2, �1, 2�
4. A�0, 5, �3�, B�5, �4, �2�, C��4, 1, 5�
6.
y
x
12
4
3
−2
−3
−4
−5
321−2
−4
−3−4−5
(3, 0, 0)
(−3, −2, −1)
z
1. A��1, 4, 3�, B�1, 3, �2�, C��3, 0, �2�
3. A��2, �1, 4�, B�3, �2, 0�, C��2, 2, �3�
5.
y
x
5432−2
−2
−3
1
3
5
2
4
1
2
3
4
5
(2, 1, 3)( 1, 2, 1)−
z 7.
y
x
12
4
1
2
3
−2
−3
−4
−5
1 2 3−2
−3−4
−3−4−5
(3, −1, 0)
(−4, 2, 2)z
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Section 10.1 The Three-Dimensional Coordinate System 889
8.
y
x
5 632−3 1
−3−4
−4
−4
−5
65
12345
(0, 4, 3)−
(4, 0, 4)
z
12. x � 6, y � �1, z � �1 ⇒ �6, �1, �1�
14. x � 0, y � 2, z � 8 ⇒ �0, 2, 8�
16. Octant VI
18. Octants III, IV, VII, or VIII 20. Octants I, II, VII, or VIII
9.
y
x
12
45
1
2
3
4
5
2
3
−
−
−3
(3, −2, 5)
z
32
, 4 ,−2( (
1 2 3 5
10.
y
x
12
34
6
2
1
1
−3
−2
−4
−5
−6
−3−5−6−7
z
(5, −2, 2)
(5, −2, −2)
11. ��3, 3, 4�z � 4:y � 3,x � �3,
13. �10, 0, 0�x � 10:y � z � 0,
15. Octant IV 17. Octants I, II, III, IV(above the xy-plane)
19. Octants II, IV, VI, VIII
21.
� 3�21 � 13.748
� �189
� �16 � 4 � 169
� �42 � 22 � 132
d � ��7 � 3�2 � �4 � 2�2 � �8 � ��5��2
23.
� 10.677
� �114
� �49 � 16 � 49
� �72 � 42 � 72
d � ��6 � ��1��2 � �0 � 4�2 � ��9 � ��2��2
22.
� �13
� �4 � 9
d � ��4 � 2�2 � �1 � 1�2 � �9 � 6�2
24.
� 5
� �25
� �9 � 16
d � ��1 � ��2��2 � �1 � ��3��2 � ��7 � ��7��2
26.
� �113
� �4 � 100 � 9
d � ��2 � 0�2 � ��4 � 6�2 � �0 � ��3��225.
� �110 � 10.488
� �1 � 9 � 100
d � ��1 � 0�2 � �0 � ��3��2 � ��10 � 0�2
27.
d1 2 � d3
2 � 20 � 9 � 29 � d2 2
d3 � ���2 � 0�2 � �5 � 4�2 � �2 � 0�2 � 3
d2 � ��0 � ��2��2 � �0 � 5�2 � �2 � 2�2 � �29
d1 � ��0 � 0�2 � �0 � 4�2 � �2 � 0�2 � �20 � 2�5
28.
d1 2 � d3
2 � 56 � 6 � 62 � d2 2
d3 � ���4 � ��2��2 � �4 � 5�2 � �1 � 0�2 � �6
d2 � �2 � ��4��2 � ��1 � 4�2 � �2 � 1�2 � �62
d1 � ��2 � ��2��2 � ��1 � 5�2 � �2 � 0�2 � �56 � 2�14
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890 Chapter 10 Analytic Geometry in Three Dimensions
29.
d1 2 � d3
2 � 9 � 36 � 45 � d2 2
d3 � ��2 � 0�2 � ��4 � 0�2 � �4 � 0�2 � �36 � 6
d2 � ��2 � 2�2 � ��4 � 2�2 � �4 � 1�2 � �45 � 3�5
d1 � ��2 � 0�2 � �2 � 0�2 � �1 � 0�2 � �9 � 3
30.
d1 2 � d3
2 � 9 � 4 � 13 � d2 2
d3 � ��1 � 1�2 � �0 � 0�2 � �3 � 1�2 � 2
d2 � ��1 � 1�2 � �3 � 0�2 � �1 � 3�2 � �13
d1 � ��1 � 1�2 � �3 � 0�2 � �1 � 1�2 � �9 � 3
33.
Since the triangle is isosceles.d1 � d3,
d3 � ��4 � 2�2 � ��1 � 3�2 � ��2 � 2�2 � �36 � 6
d2 � ��8 � 2�2 � �1 � 3�2 � �2 � 2�2 � �40 � 2�10
d1 � ��8 � 4�2 � �1 � 1�2 � �2 � 2�2 � �36 � 6
34.
Right triangle
d1 2 � d3
2 � 9 � 36 � 45 � d2 2
d3 � ��3 � 1�2 � ��6 � 2�2 � �3 � 1�2 � �36 � 6
d2 � ��3 � 3�2 � �0 � 6�2 � �0 � 3�2 � �45 � 3�5
d1 � ��3 � 1�2 � �0 � 2�2 � �0 � 1�2 � �9 � 3
31.
Isosceles triangled1 � d3,
d3 � ���1 � 1�2 � �1 � 3�2 � �2 � 2�2 � �4 � 16 � 16 � �36 � 6
d2 � ��5 � 1�2 � ��1 � 1�2 � �2 � 2�2 � �36 � 4 � �40 � 2�10
d1 � ��5 � 1�2 � ��1 � 3�2 � �2 � 2�2 � �16 � 4 � 16 � �36 � 6
32.
Isosceles triangled1 � d3 � 3,
d3 � ��3 � 5�2 � �5 � 3�2 � �3 � 4�2 � �4 � 4 � 1 � �9 � 3
d2 � ��3 � 7�2 � �5 � 1�2 � �3 � 3�2 � �16 � 16 � �32 � 4�2
d1 � ��7 � 5�2 � �1 � 3�2 � �3 � 4�2 � �4 � 4 � 1 � �9 � 3
35. Midpoint: �3 � 32
, �6 � 4
2,
10 � 42 � � �0, �1, 7�
37. Midpoint: �6 � 42
, �2 � 2
2,
5 � 62 � � �1, 0,
112 �
36. Midpoint: ��1 � 32
, 5 � 7
2,
�3 � 12 � � �1, 6, �2�
38. Midpoint: ��3 � 62
, 5 � 4
2,
5 � 82 � � ��
92
, 92
, 132 �
40. Midpoint: �9 � 92
, �5 � 2
2,
1 � 42 � � �9, �
72
, �32�
42. �x � 3�2 � �y � 4�2 � �z � 3�2 � 4
39. Midpoint: ��2 � 72
, 8 � 4
2,
10 � 22 � � �5
2, 2, 6�
41. �x � 3�2 � � y � 2�2 � �z � 4�2 � 16
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Section 10.1 The Three-Dimensional Coordinate System 891
43. �x � 1�2 � �y � 2�2 � z2 � 3 44. x2 � �y � 1�2 � �z � 3�2 � 5
46. �x � 2�2 � �y � 1�2 � �z � 8�2 � 3645.
x2 � � y � 4�2 � �z � 3�2 � 9
�x � 0�2 � � y � 4�2 � �z � 3�2 � 32
47.
�x � 3�2 � �y � 7�2 � �z � 5�2 � 52 � 25
Radius �Diameter
2� 5 48.
�x � 0�2 � � y � 5�2 � �z � 9�2 � 42 � 16
Radius �Diameter
2� 4
50. Center:
Radius:
Sphere: �x �1
2�2
� �y � 1�2 � �z � 4�2 �61
4
��2 �1
2�2
� ��2 � 1�2 � �2 � 4�2 ��9
4� 9 � 4 ��61
4
�2 � 1
2,
�2 � 4
2,
2 � 6
2 � � �1
2, 1, 4�
52.
Center:
Radius: 4
�0, 4, 0�
x2 � �y � 4�2 � z2 � 16
x2 � y2 � 8y � 16 � z2 � 16
49. Center:
Radius:
Sphere: �x �3
2�2
� � y � 0�2 � �z � 3�2 �45
4
��3 �3
2�2
� �0 � 0�2 � �0 � 3�2 ��9
4� 9 ��45
4
�3 � 0
2,
0 � 0
2,
0 � 6
2 � � �3
2, 0, 3�
51.
Center:
Radius: 52
�52, 0, 0�
�x �52�2
� y2 � z2 �254
�x2 � 5x �254 � � y2 � z2 �
254
53.
Center:
Radius: �5
�2, �1, 0�
�x � 2�2 � �y � 1�2 � z2 � 5
�x2 � 4x � 4� � �y2 � 2y � 1� � z2 � 4 � 1
54.
Center:
Radius:�32
�12
, 12
, 12�
�x �12�
2� �y �
12�
2� �z �
12�
2�
34
�x2 � x �14� � �y2 � y �
14� � �z2 � z �
14� �
14
�14
�14
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892 Chapter 10 Analytic Geometry in Three Dimensions
55.
Center:
Radius: 2
�2, �1, 3�
�x � 2�2 � �y � 1�2 � �z � 3�2 � 4
�x2 � 4x � 4� � �y2 � 2y � 1� � �z2 � 6z � 9� � �10 � 4 � 1 � 9
57.
Center:
Radius: 1
��2, 0, 4�
�x � 2�2 � y2 � �z � 4�2 � 1
�x2 � 4x � 4� � y2 � �z2 � 8z � 16� � �19 � 4 � 16
56.
Center:
Radius: 2
�3, �2, 0�
�x � 3�2 � �y � 2�2 � z2 � 4
�x2 � 6x � 9� � �y2 � 4y � 4� � z2 � �9 � 9 � 4
58.
Center:
Radius: �12 � 2�3
�0, 4, 3�
x2 � � y � 4�2 � �z � 3�2 � 12
x2 � � y2 � 8y � 16� � �z2 � 6z � 9� � �13 � 16 � 9
60.
Center:
Radius: 1
�12, 32, 1�
�x �12�2
� �y �32�2
� �z � 1�2 � 1
�x2 � x �14� � �y2 � 3y �
94� � �z2 � 2z � 1� � �
52 �
14 �
94 � 1
x2 � y2 � z2 � x � 3y � 2z � �52
61.
Center:
Radius:�21
2
�1, �2, 0�
�x � 1�2 � �y � 2�2 � z2 �214
4�x2 � 2x � 1� � 4�y2 � 4y � 4� � 4z2 � 4 � 16 � 1
59.
Center:
Radius: 3
�1, 13, 4� �x � 1�2 � �y �
13�2
� �z � 4�2 � 9
�x2 � 2x � 1� � �y2 � 23y � 1
9� � �z2 � 8z � 16� � � 739 � 1 � 1
9 � 16
x2 � y2 � z2 � 2x � 23y � 8z � � 73
9
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Section 10.1 The Three-Dimensional Coordinate System 893
62.
Center:
Radius: 2�7
�1, �2, �3�
�x � 1�2 � �y � 2�2 � �z � 3�2 � 28 � �2�7�2
9�x � 1�2 � 9�y � 2�2 � 9�z � 3�2 � 252
9�x2 � 2x � 1� � 9�y2 � 4y � 4� � 9�z2 � 6z � 9� � 9 � 36 � 81 � 126
64.
Center:
Radius: 3
�12
, 4, �1�
�x �12�
2
� �y � 4�2 � �z � 1�2 � 9
x2 � x �14
� y2 � 8y � 16 � z2 � 2z � 1 ��33
4�
14
� 16 � 1
63.
Center:
Radius: 1
�13, �1, 0�
�x �13�2
� �y � 1�2 � z2 � 1
x2 �23 x �
19 � y2 � 2y � 1 � z2 � �
19 �
19 � 1
9x2 � 6x � 9y2 � 18y � 9z2 � �1
65.
y
x
2
−2
2
2
(1, 0, 0)
( 1) + = 36x z− 2 2
z
xz-trace � y � 0�: �x � 1�2 � z2 � 36, Circle
67.
Circle
y
x
2
2 2
( 3) + = 5y z− 2 2
( 2, 3, 0)−
z
yz-trace �x � 0�: � y � 3�2 � z2 � 9 � 4 � 5,
66.
(0, 3, 0)−
2
−6
4
6
86
2
−6−8
−4−2
−10
x
y
( + 3) + = 25y z2 2
z
yz-trace �x � 0�: � y � 3�2 � z2 � 25, Circle
68.
2
−2−4 −42
−4
−6
4
46x
y
(0, 1, 1)−
x y2 2+ ( 1) = 3−
z
xy-trace �z � 0�: x2 � � y � 1�2 � 3, Circle
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894 Chapter 10 Analytic Geometry in Three Dimensions
71.
yx
2 233
4455
5
6
z
73.
7
43
−3
65
7
z
x y
74.
x
y
21
1
−1
−1
−2
−1−2
−3
−3
−4
z
69.
y2 � �z � 2�2 � 3yz � trace: x � 0:
z
xy
23
4
−1−2
32
−2
3
2
5y2 + (z − 2)2 = 3
�x � 1�2 � y2 � �z � 2�2 � 4
�x2 � 2x � 1� � y2 � �z2 � 4z � 4� � �1 � 1 � 4 70.
x2 � �z � 3�2 � 21xz � trace: y � 0:
−6
−4
4
6
12
10
4
810
86
−6−4
xy
z
x2 + (z − 3)2 = 21
x2 � �y � 2�2 � �z � 3�2 � 25
x2 � �y2 � 4y � 4� � �z2 � 6z � 9� � 12 � 4 � 9
72.
x
y2
4
−2−2
−4−4
−62
z
z2 � 4 � ��5 � x2 � y2 � 6y
z1 � 4 � ��5 � x2 � y2 � 6y
x2 � y2 � 6y � �z � 4�2 � �5
x2 � y2 � 6y � �z2 � 8z � 16� � �21 � 16
75. The length of each side is 3.Thus, �x, y, z� � �3, 3, 3�.
77.
x2 � y2 � z2 � �1652 �2
d � 165 ⇒ r �1652 � 82.5
79. False. x is the directed distance from the yz-plane to P.
76. x � 4, y � 4, z � 8, �4, 4, 8�
78. (a)
(b) Assume the north and south poles are on the -axis. Lines of longitude that run north–south are traces of planes containing the -axis. These shapes are circles of radius 3963 miles.
(c) Latitudes are traces of planes perpendicular to the -axis. These shapes are circles.z
zz
x2 � y2 � z2 � 39632
80. False. The trace could be a single point, or empty.
82. It is a plane.81. In the xy-plane, the z-coordinate is 0.In the xz-plane, the y-coordinate is 0.In the yz-plane, the x-coordinate is 0.
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Section 10.1 The Three-Dimensional Coordinate System 895
84. The trace will be a line in the xy-plane (unless theplane is the xy-plane).
83. The trace is a circle, or a single point.
85.
Similarly for
�x2, y2, z2� � �2xm � x1, 2ym � y1, 2zm � z1�.
y2 and z2,
xm �x2 � x1
2 ⇒ x2 � 2xm � x1
87.
v � �32
±�17
2
v �32
� ±�17
2
�v �32�
2
�174
v2 � 3v �94
� 2 �94
89.
x �52
±�52
x �52
� ±�52
�x �52�
2
�54
x2 � 5x �254
� �5 �254
91.
y � �12
±�10
2
y �12
� ±�10
2
�y �12�
2
�104
y2 � y �14
�94
�14
4y2 � 4y � 9
86.
�7, 16, 12�
z2 � 2zm � z1 � 2�7� � 2 � 12
y2 � 2ym � y1 � 2�8� � 0 � 16
x2 � 2xm � x1 � 2�5� � 3 � 7
88.
z �72
±52�5
z �72
� ±5�5
2
�z �72�
2
�1254
z2 � 7z �494
� 19 �494
90.
x ��32
±�13
2
x �32
� ±�13
2
�x �32�
2
�134
x2 � 3x �94
� 1 �94
92.
x ��54
±�89
4
x �54
� ±�89
4
�x �54�
2
�8916
x2 �52
x �2516
� 4 �2516
93. Quadrant IV
� � �45� or 315�
tan � � �33
� �1 ⇒
� 3�2
� �18
v � �32 � ��3�2
v � 3i � 3j, 95. Quadrant I
tan � �54
⇒ � � 51.34�
v � �16 � 25 � �41
v � 4i � 5j,
97.
� �7
� �4�3� � 1�5�
u � v � �4, 1� � 3, 5�
94. Quadrant II
tan � �2
�1 ⇒ � � 116.6�
v � �12 � 22 � �5
v � �1, 2�,
96. Quadrant IV
tan � ��710
⇒ � � 325.0�
v � �100 � 49 � �149
v � 10, �7�, 98.
� 2
� 2 � 0
u � v � �1, 0� � �2, �6�
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896 Chapter 10 Analytic Geometry in Three Dimensions
99.
1 2 6 15 31
First differences: 1 4 9 16
Second differences: 3 5 7
Neither model
a4 � 15 � 42 � 31
a3 � 6 � 32 � 15
a2 � 2 � 22 � 6
a1 � 1 � 12 � 2
a0 � 1, an � an�1 � n2
100.
0
First differences:
Second differences: 0 0 0
Linear model
�1�1�1�1
�4�3�2�1
a4 � �4
a3 � �3
a2 � �1 � 1 � �2
a1 � 0 � 1 � �1
a0 � 0, an � an�1 � 1
102.
4 0
First differences:
Second differences:
Quadratic model
�2�2�2
�10�8�6�4
�24�14�6
a5 � �14 � 2�5� � �24
a4 � �6 � 2�4� � �14
a3 � 0 � 2�3� � �6
a2 � 4 � 2�2� � 0
a1 � 4, an � an�1 � 2n
101.
2 5 8 11
First differences: 3 3 3 3
Second differences: 0 0 0
Linear model
�1
a5 � 8 � 3 � 11
a4 � 5 � 3 � 8
a3 � 2 � 3 � 5
a2 � �1 � 3 � 2
a1 � �1, an � an�1 � 3
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Section 10.2 Vectors in Space 897
104. �x � 3�2 � �y � 6�2 � 81
106.
�x � 2�2 � �20�y � 5�
�x � 2�2 � 4��5��y � 5�
�h, k� � ��2, 5�p � �5, �x � h�2 � 4p� y � k�,
103. �x � 5�2 � �y � 1�2 � 49
105.
�y � 1�2 � �12�x � 4�
�y � 1�2 � 4��3��x � 4�
�y � 1�2 � 4p�x � 4�, p � �3
108. Center:
Vertical major axis length 9
�x � 0�2
�45�4� �� y � 3�2
�81�4� � 1
b2 � a2 � c2 �814
� 9 �454
⇒c � 3
a �92
⇒
�0, 3�
110. Center: vertical transverse axis
�y � 5�2
16�
�x � 3�2
9� 1
a � 4, c � 5, b2 � c2 � a2 � 25 � 16 � 9
�3, 5�,
107. center: horizontal major axis
�x � 3�2
9�
�y � 3�2
4� 1
�3, 3�,a � 3, b � 2,
109. Center: horizontal transverse axis
�x � 6�2
4�
y2
32� 1
a � 2, c � 6, b2 � c2 � a2 � 36 � 4 � 32
�6, 0�,
Section 10.2 Vectors in Space
■ Vectors in space have many of the same properties as vectors in the plane.
■ The dot product of two vectors and in space is
■ Two nonzero vectors u and v are said to be parallel if there is some scalar c such that
■ You should be able to use vectors to solve real life problems.
u � cv.
u1v1 � u2v2 � u3v3.u � v �v � �v1, v2, v3�u � �u1, u2, u3�v � �v1, v2, v3�
Vocabulary Check
1. zero 2. 3. component form
4. orthogonal 5. parallel
v � v1i � v2 j � v3k
1.
y
x
3
2
1
1
−1
−3
−2−3
12
3
23
z
(−2, 3, 1)
v � �0 � 2, 3 � 0, 2 � 1� � ��2, 3, 1� 2. (a)
(b)
y
x
24
68
4
2
6
8
642 8−4
−6
−8
−4
−6−8
z
(−1, 6, −8)
� ��1, 6, �8�
v � �0 � 1, 4 � ��2�, �4 � 4�
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898 Chapter 10 Analytic Geometry in Three Dimensions
3. (a)
(b)
y
x
3
12
34
−2
−3−2
−1
−4
−3−4
23
4
2
−4
1
−2
−3
z
(0, 0, −4)
v � �1 � 1, 4 � 4, 0 � 4� � �0, 0, �4� 4. (a)
(b)
y
x
12
2
3
4
321 4−2
−3−4
−5−6
−3
−2
−4
−3−4
z
(−4, 0, 0)
v � �0 � 4, �2 � ��2�, 1 � 1� � ��4, 0, 0�
5. (a)
(b)
(c)v
�v ��
1
3�11�7, �5, 5� �
�1133
�7, �5, 5�
� 3�11
� �99
� �49 � 25 � 25
�v � � �72 � ��5�2 � 52
� �7, �5, 5�
v � �1 � ��6�, �1 � 4, 3 � ��2�� 6. (a)
(b)
(c) Unit vector:1
�67�7, �3, �3� �
�6767
�7, �3, �3�
�v� � �49 � 9 � 9 � �67
v � �0 � 7, 0 � 3, 2 � 5� � �7, �3, �3�
7. (a)
(b)
(c) Unit vector:1
2�2�2, 2, 0� � ��2
2 , �22
, 0�v� � �22 � 22 � 02 � �8 � 2�2
v � �1 � ��1�, 4 � 2, �4 � ��4�� � �2, 2, 0� 8. (a)
(b)
(c) Unit vector:13
�0, 3, 0� � �0, 1, 0�
�v� � �02 � 32 � 02 � �9 � 3
v � �0 � 0, 2 � ��1�, 1 � 1� � �0, 3, 0�
9. (a)
y
x
12
34
1
2
3
4
5
6
−2
31 4−2−3−4
z
⟨2, 2, 6⟩
(b)
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−4
−3−4
z
⟨−1, −1, −3⟩
(c)
y
x
12
34
2
1
3
4
5
32 4−2
−3
−2
−3−4
z
32
, 32
, 92
(d)
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z
⟨0, 0, 0⟩
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Section 10.2 Vectors in Space 899
10.
(a)
(c)
y
x
1
2
2
1
21−1
−2
−1
−2
−2
z
12
− , 1, 1
12
v � ��12
, 1, 1
y
x
2
1
3
4
21
−3−4
−3
−2
−4
−4 −3− 5 −6
z
⟨1, −2, −2⟩
�v � �1, �2, �2�
v � ��1, 2, 2�
(b)
(d)
y
x
12
34
2
1
3
4
5
6
321 4 5−2
−3−4
−2
−3
z52
− , 5, 5
52
v � ��52
, 5, 5
y
x
12
34
2
1
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z ⟨−2, 4, 4⟩
2v � ��2, 4, 4�
11.
(a)
(c)
y
x
12
34
56
1
2
3
321 4 6−2
−3
−2
−4
−5
z
5, 5, − 52
52
v � 5i � 5j �52
k
y
x
12
34
2
1
321−2
−3−4
−3
−4
−6
−5
−3−4
z
⟨4, 4, −2⟩
2v � 4i � 4j � 2k
v � 2i � 2j � k
(b)
(d)
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z
⟨0, 0, 0⟩
0v � 0
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z
⟨−2, −2, 1⟩
�v � �2i � 2j � k
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900 Chapter 10 Analytic Geometry in Three Dimensions
12.
(a)
(c)
y
x
1
2
2
1
21
−2
−1
−2
−2
z
12 , −1, 1
2
12v �
12i � j �
12k
y
x
2
68
4
2
6
8
10
42
−6
−4
−6−12 −4
−6−8
z
⟨4, −8, 4⟩
4v � 4i � 8j � 4k
v � i � 2j � k
(b)
(d)
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z
⟨0, 0, 0⟩
0v � 0
y
x
12
34
2
3
4
1 6−2
−3−4
−3
−2
−4
z
⟨−2, 4, −2⟩
�2v � �2i � 4j � 2k
13. z � u � 2v � ��1, 3, 2� � 2�1, �2, �2� � ��3, 7, 6�
14. z � 7��1, 3, 2� � �1, �2, �2� �15 �5, 0, �5� � ��7, 19, 13�
17. z � 2��1, 3, 2� � 3�1, �2, �2� �12 �5, 0, �5� � ��
52, 12, 15
2 �
18. z � 3�5, 0, �5� � 2�1, �2, �2� � ��1, 3, 2� � �12, 7, �9�
19.
z � �112 , �5
4, �6�4z � 4�5, 0, �5� � ��1, 3, 2� � �1, �2, �2� � �22, �5, �24�
20. � �5, 0, �5� � ��1, 3, 2� � 2�1, �2, �2� � �4, 1, �3�z � w � u � 2v
15. 2z � 4u � w ⇒ z �12�4u � w� �
12�4��1, 3, 2� � �5, 0, �5�� � �1
2, 6, 32�
16. z � �u � v � ���1, 3, 2� � �1, �2, �2� � �0, �1, 0�
21.
� �49 � 64 � 49 � �162 � 9�2
�v � � ��7, 8, 7�� 22. �v� � ���2�2 � 02 � ��5�2 � �4 � 25 � �29
24. �v� � ���1�2 � 02 � 32 � �10
26. �v� � �12 � 32 � ��1�2 � �11
23. �v� � �12 � ��2�2 � 42 � �21
25. �v� � �22 � ��4�2 � 12 � �21
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Section 10.2 Vectors in Space 901
27.
� �16 � 9 � 49 � �74
�v � � �42 � ��3�2 � ��7�2
31.
(a)
(b) �113�5i � 12k�
113�5i � 12k�
�u � � �52 � ��12�2 � �169 � 13 32.
(a)
(b) �15�3i � 4k� � �
35i �
45k
15�3i � 4k� �
35i �
45k
�u � � �32 � ��4�2 � �25 � 5
28. �v� � �22 � ��1�2 � 62 � �41
29.
�v � � �0 � 32 � ��5�2 � �34
v � �1 � 1, 0 � ��3�, �1 � 4� � �0, 3, �5� 30.
�v� � �1 � 9 � 4 � �14
v � �1 � 0, 2 � ��1�, �2 � 0� � �1, 3, �2�
33. (a)
(b) �1
�74�8i � 3j � k� � �
�74
74�8, 3, �1�
�1
�74�8i � 3j � k� �
�7474
�8, 3, �1�
u
�u��
�8, 3, �1��74
34. (a)
(b)�1
�134��3i � 5j � 10k�
u�u�
���3, 5, 10��134
�1
�134��3i � 5j � 10k�
35. � ��26, 0, 48�� ��6, 18, 24� � ��20, �18, 24�6u � 4v � 6��1, 3, 4� � 4�5, 4.5, �6�
36. 2u �52 v � 2��1, 3, 4� �
52�5, 4.5, �6� � �21
2 , 694 , �7�
37.
�u � v� ��42 � 7.52 � ��2�2 �12�305 8.73
u � v � ��1, 3, 4� � �5, 4.5, �6� � �4, 7.5, �2�
38.v
�v��
�5, 4.5, �6��25 � 20.25 � 36
��5, 4.5, �6�
5�13�2� � 2
�13,
9
5�13,
�12
5�13 �0.5547, 0.4992, �0.6656�
39.
� 8 � 20 � 8 � �4
u � v � �4, 4, �1� � �2, �5, �8�
41.
� 18 � 15 � 3 � 0
u � v � �2, �5, 3� � �9, 3, �1�
40. u � v � 3�4� � ��1���10� � 6�1� � 28
42. u � v � 0�6� � 3��4� � ��6���2� � 0
43. cos � �u � v
�u� �v��
�8
�8�25 ⇒ � 124.45� 44. cos � �
u � v�u� �v�
�5
�10�6 ⇒ � 49.80�
45. cos � �u � v
�u� �v��
�120�1700�73
⇒ � 109.92� 46. cos � �u � v
�u� �v��
100�464�125
⇒ � 65.47�
47. �32 �8, �4, �10� � ��12, 6, 15� ⇒ parallel 48. and
u � cv ⇒ neither
u � v � �2 � 3 � 5 � �10 � 0
49. u � v � 3 � 5 � 2 � 0 ⇒ orthogonal
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902 Chapter 10 Analytic Geometry in Three Dimensions
50. �8u � �8��1, 12, �1� � �8, �4, 8� � v ⇒ parallel
51.
Neither parallel nor orthogonal
u � v � �2 � 6 � 0
u � cv 52.
Neither parallel nor orthogonal
u � v � 4 � 0
u � cv
53.
Orthogonal
u � v � �4 � 3 � 1 � 0 54.
Orthogonal
u � v � �2 � 3 � 1 � 0
55.
Since u and v are not parallel, the points are notcollinear.
u � �4 � 7, 5 � 3, 3 � ��1�� � ��3, 2, 4�
v � �7 � 5, 3 � 4, �1 � 1� � �2, �1, �2� 56.
Since the points are collinear.u � �2v,
u � �0 � ��4�, 6 � 8, 7 � 1� � �4, �2, 6�
v � ��4 � ��2�, 8 � 7, 1 � 4� � ��2, 1, �3�
57.
Since the points are collinear.u � �2v,
u � �3 � ��1�, 4 � 2, �1 � 5� � �4, 2, �6�
v � ��1 � 1, 2 � 3, 5 � 2� � ��2, �1, 3� 58.
Since u and v are not parallel, the points are notcollinear.
u � ��2 � ��1�, 6 � 5, 7 � 6� � ��1, 1, 1�
v � ��1 � 0, 5 � 4, 6 � 4� � ��1, 1, 2�
59. The vector joining and isperpendicular to the vector joining
and :
The triangle is a right triangle.
�1, 2, 0� � ��2, 1, 0� � �2 � 2 � 0
�0, 0, 0���2, 1, 0���2, 1, 0�
�0, 0, 0��1, 2, 0��1, 2, 0� 60. Consider the vector joining andand the vector joining
and :
The triangle has an obtuse angle.
Obtuse triangle
��3, 0, 0� � �1, 2, 3� � �3 < 0
�0, 0, 0��1, 2, 3��1, 2, 3���3, 0, 0�
�0, 0, 0���3, 0, 0�
61. The three sides of the triangle are given by the vectors:
The triangle has three acute angles.
Acute triangle
v � w � 16 > 0
u � w � 10 > 0
u � v � 34 > 0
w � ��1, 1, �2�
v � ��3, 5, �4�
u � ��2, 4, �2�
62. Consider the vector joining and , and the vector joining
and :
The triangle has an obtuse angle.
Obtuse triangle
�5, 1, �9� � ��3, 12, 5� � �48 < 0
��1, 5, 8��4, 6, �1��5, 1, �9��2, �7, 3�
��1, 5, 8���3, 12, 5�
63.
Terminal point is �3, 1, 7�.2
�4
7
�
�
�
q1 � 1
q2 � 5
q3
� ⇒ q1 � 3
q2 � 1
q3 � 7� ⇒
v � �2, �4, 7� � �q1 � 1, q2 � 5, q3 � 0� ⇒
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Section 10.2 Vectors in Space 903
64. �4, �1, �1� � �x � 6, y � 4, z � 3� ⇒ �x, y, z� � �10, �5, 2�
65.
Terminal point: �6, 52, �74�
�14 � q3 �
32 ⇒ q3 � �
74
32 � q2 � 1 ⇒ q2 �52
4 � q1 � 2 ⇒ q1 � 6
v � �4, 32, �14� � �q1 � 2, q2 � 1, q3 �
32�
66. �52, �1
2, 4� � �x � 3, y � 2, z �12� ⇒ �x, y, z� � �11
2 , 32, 72�
67.
c � ±3
�14� ±
3�14
14
�cu� ��c2 � 4c2 � 9c2 � �c��14 � 3 ⇒
cu � ci � 2cj � 3ck 68.
⇒ �c� �12�24
�6�6
� �6 ⇒ c � ±�6
�c u� � �c� �u� � �c��4 � 4 � 16 � �c��24 � 12
69.
Since lies in the plane, Since makesan angle of Finally, impliesthat Thus,and or and
and v � �0, 2�2, �2�2 �.q3 � �2�2q2 � 2�2v � �0, 2�2, 2�2�,
q2 � q3 � 2�2q22 � q3
2 � 16.�v� � 4�q2���q3�.45�,
vq1 � 0.yz-v
v � �q1, q2, q3� 70. lies in xz-plane
or
v � 10��sin 60�, 0, cos 60�� � ��5�3, 0, 5�v � 10�sin 60�, 0, cos 60�� � �5�3, 0, 5�,
⇒ y � 0.v
71.
Let and be the tension on each wire. Since there exists a constant c such that
The total force is the vertical component satisfies
Hence,
�F1� � �F2� � �F3� 10.91 pounds.
F � ��10
�7,
10
�21, �10
F2 � � 10
�7,
10
�21, �10
F1 � �0, �20
�21, �10
�10 � �12�21c ⇒ c �5
6�21.
�k��30k � F1 � F2 � F3 ⇒
F3 � c ��12�3, 12, �12�21�.
F2 � c �12�3, 12, �12�21�F1 � c �0, �24, �12�21�
�F1� � �F2� � �F3�,F3F1, F2,
PQ3
\
� ��12�3, 12, �12�21�PQ2
\
� �12�3, 12, �12�21�PQ1
\
� �0, �24, �12�21�
P � �0, 0, 55�
Q3 � ��20.8, 12, 0�
Q2 � �20.8, 12, 0�
Q1 � �0, �24, 0�
x y
Q1 = (0, 24, 0)−
Q3 = 24 , 0−
Q2 = 24 , 0
(
(
)
)
,
,
P = (0, 0, 12 21)
3
3
1
1
2
2
2
2
10
−10 −10
40
30
10
z
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904 Chapter 10 Analytic Geometry in Three Dimensions
72.
Thus
Solving this system yields
Thus,
N
N
N.�F3� 226.521
�F2� 157.909
�F1� 202.919
C3 ��112
69.C1 �
�10469
, C2 ��2823
,
115C1 � 115C2 � 115C3 � �500.
70C1 � 65C3 � 0
�60C2 � 45C3 � 0
F1 � F2 � F3 � �0, 0, �500�.
F3 � C3�45, �65, 115�AD\
� �45, �65, 115�,
F2 � C2��60, 0, 115�AC\
� ��60, 0, 115�,
F1 � C1�0, 70, 115�AB\
� �0, 70, 115�,
73. True. � � 90�⇒cos � � 0 74. True
75. (a)
(c)
Hence, a � b � 1.
1 � b
2 � a � b
1 � a
w � �1, 2, 1� � a�1, 1, 0� � b�0, 1, 1�
y
x
3
2
1
−1
−2−3
−2−3
23
23
z
uv
(b)
(d)
Impossible
3 � b
2 � a � b
1 � a
w � �1, 2, 3� � a�1, 1, 0� � b�0, 1, 1�
0 � �a, a � b, b� ⇒ a � b � 0
w � au � bv � a�1, 1, 0� � b�0, 1, 1�
76. This set is a sphere.
�x � x1�2 � �y � y1�2 � �z � z1�2 � 16
77. If then and the angle betweenand is obtuse, 180� > � > 90�.vu
cos � < 0u � v < 0,
78. Let and
Then
The endpoints of these three vectors are collinear, as indicated in the figure. So, the figure is a line.
su � tv � �su1 � tv1, su2 � tv2, su3 � tv3�.
u � tv � �u1 � tv1, u2 � tv2, u3 � tv3�, and
tv � �tv1, tv2, tv3�,
u
su
v tv
u v+ t
s tu v+
u � �u1, u2, u3�.v � �v1, v2, v3�
©H
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.
Section 10.3 The Cross Product of Two Vectors 905
79. (a)
(b)
� 3t � 1 y � 3�t � 1� � 2
x � t � 1
y � 3t � 2
x � t 80. (a)
(b) x � t � 1, y �2
t � 1
x � t, y �2t
82. (a)
(b) x � t � 1, y � 4�t � 1�3
x � t, y � 4t381. (a)
(b)
� t2 � 2t � 7 y � �t � 1�2 � 8
x � t � 1
y � t2 � 8
x � t
Section 10.3 The Cross Product of Two Vectors
■ The cross product of two vectors and is given by
■ The cross product satisfies the following algebraic properties.
(a)
(b)
(c)
(d)
(e)
(f)
■ The following geometric properties of the cross product are valid, where is the angle between the vectors u and v:
(a) is orthogonal to both u and v.
(b)
(c) if and only if u and v are scalar multiples.
(d) is the area of the parallelogram having u and v as sides.
■ The absolute value of the triple scalar product is the volume of the parallelepiped having u, v, and w as sides.
u � �v � w� � � u1
v1
w1
u2
v2
w2
u3
v3
w3��u � v�u � v � 0
�u � v� � �u� �v� sin �
u � v
�
u � �v � w� � �u � v� � w
u � u � 0
u � 0 � 0 � u � 0
c�u � v� � �cu� � v � u � �cv�u � �v � w� � �u � v� � �u � w�u � v � ��v � u�
� � i u1
v1
j u2
v2
k
u3
v3�.u � v � �u2v3 � u3v2�i � �u1v3 � u3v1�j � �u1v2 � u2v1�k
v � v1i � v2 j � v3ku � u1i � u2 j � u3k
Vocabulary Check
1. cross product 2. 0
3. 4. triple scalar product�u� �v� sin �
©H
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.
906 Chapter 10 Analytic Geometry in Three Dimensions
1.
y
x
1
−1
−2
2
−1
−21
−2
−1
2
(0, 0, 1)−
z
j � i � � i0
1
j1
0
k0
0� � �k 2.
y
x
1
2
−2
−1
2
−21
−2
−1
2
( 1, 0, 0)−
z
k � j � � i00
j01
k10� � �i
3.
y
x
1
2
−2
2
−1
−21
−2
−1
2
(0, 1, 0)−
z
i � k � � i1
0
j0
0
k0
1� � �j 4.
y
x
1
2
−2
−1
2
−1
−21
−2
−1
2
(0, 1, 0)
z
k � i � � i0
1
j0
0
k1
0� � j
5. u � v � � i1
0
j�1
1
k0
�1� � i � j � k � �1, 1, 1�
6. u � v � � i�1
1
j1
0
k0
�1� � �i � j � k � ��1, �1, �1�
8. u � v � � i2
1
j�3
�2
k1
1� � �i � j � k � ��1, �1, �1�
7.
�u � v� � v � �3, �3, �3� � �0, �1, 1� � 0
�u � v� � u � �3, �3, �3� � �3, �2, 5� � 0
u � v � � i30
j�2�1
k51� � �3, �3, �3�
9.
�u � v� � v � �0, 42, 0� � �7, 0, 0� � 0
�u � v� � u � �0, 42, 0� � ��10, 0, 6� � 0
u � v � � i�10
7
j00
k60� � �0, 42, 0� 10. u � v � � i
�5
2
j5
2
k11
3� � ��7, 37, �20� ©H
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.
Section 10.3 The Cross Product of Two Vectors 907
11.
� �7i � 13j � 16k
u � v � � i61
j23
k1
�2� � ��7, 13, 16�
12. u � v � � i
112
j32
�34
k
�5214� � ��
32, �3
2, �32� � �
32 i �
32 j �
32 k
15. u � v � � i12
�34
j
�23
1
k
114� � �
76 i �
78 j 16. u � v � � i
25
�35
j
�14
1
k1215� � �
1120 i �
1950 j �
14k
13. u � v � � i2
�1
j4
3
k3
�2� � �17i � j � 10k 14. u � v � � i3
�2
j�2
1
k1
2� � �5i � 8j � k
17.
� �18i � 6j
u � v � � i0
�1
j03
k61� � ��18, �6, 0� 18. u � v � � i
23
0
j013
k0
�3� � 2j �2
9k
19.
� �i � 2j � k
u � v � � i�1
0
j01
k1
�2� � ��1, �2, �1�
20. u � v � � i10
j0
�1
k�2
1� � �0 � 2�i � �1 � 0�j � ��1 � 0�k � �2i � j � k
21. u � v � � i2
0
j4
�2
k3
1� � �10, �2, �4� 22. u � v � � i4
�1
j�2
5
k6
7� � ��44, �34, 18�
23. u � v � � i1
�4
j�2
2
k4
�1� � �6i � 15j � 6k 24. u � v � � i2
�1
j�1
1
k3
�4� � i � 5j � k
25. u � v � � i612
j�5
�34
k1210� � ��0.25, �0.7, �2� 26. � �
12 i � 3j � 8ku � v � � i
812
j�4
34
k2
�14�©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
908 Chapter 10 Analytic Geometry in Three Dimensions
27.
Unit vector: �166166
�9, 6, �7�
�u � v� � �166
u � v � � i1
2
j2
�3
k3
0� � �9, 6, �7� 28.
Unit vector: 1
3�6�2, 7, 1� �
�618
�2, 7, 1�
�u � v� � �54 � 3�6
u � v � � i2
1
j�1
0
k3
�2� � �2, 7, 1�
29.
��1919
�1, �3, 3�
Unit vector �u � v
�u � v ��
1
�19�i � 3j � 3k�
�u � v� � �19
u � v � � i3
0
j1
1
k0
1� � i � 3j � 3k 30.
Unit vector �u � v
�u � v�� �
67
i �37
j �27
k
�u � v� � �36 � 9 � 4 � 7
u � v � � i11
j20
k0
�3� � �6i � 3j � 2k
31.
Consider the parallel vector
��76027602
��71, �44, 25�
Unit vector �1
�7602��71, �44, 25�
�w� � �712 � 442 � 252 � �7602
��71, �44, 25� � w.
u � v � � i�3
12
j2
�34
k�5
110� � ��
71
20, �
11
5,
54 32.
��429
1287�10, 25, 56�
�1
3�429�10, 25, 56�
Unit vector �u � v
�u � v��
1
21�249�70, 175, 392�
� �189,189 � 21�429
�u � v� � �702 � 1752 � 3922
u � v � � i7
14
j�14
28
k5
�15� � 70i � 175j � 392k
33.
��22
i ��22
j
�1
�2i �
1
�2j
Unit vector �u � v
�u � v��
1
2�2�2i � 2j�
�u � v� � 2�2
u � v � � i1
1
j1
1
k�1
1� � 2i � 2j 34.
�2
3i �
2
3j �
1
3k
Unit vector �u � v
�u � v��
1
9�6i � 6j � 3k�
�u � v� � �36 � 36 � 9 � 9
u � v � � i1
2
j�2
�1
k2
�2� � 6i � 6j � 3k
35.
Area � �u � v � � �j� � 1 square unit
u � v � � i0
1
j0
0
k1
1� � j 36.
square units � �4 � 1 � 4 � 3
Area � �u � v� � �2i � j � 2k�
u � v � � i1
1
j2
0
k2
1� � 2i � j � 2k ©H
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Section 10.3 The Cross Product of Two Vectors 909
37.
� �806 square units
Area � �u � v � � �262 � ��3�2 � ��11�2
u � v � � i3
2
j4
�1
k6
5� � 26i � 3j � 11k 38.
square units � �213
Area � �u � v� � �82 � 102 � ��7�2
u � v � � i�2
1
j3
2
k2
4� � �8, 10, �7�
39.
� 14 square units
Area � �u � v� � �122 � ��6�2 � 42
u � v � � i2
0
j2
2
k�3
3� � �12, �6, 4� 40.
square units � �270 � 3�30
Area � �u � v� � ���3�2 � 62 � 152
u � v � � i4
5
j�3
0
k2
1� � ��3, 6, 15�
42. (a)
Opposites are parallel and same length. Thus, form a parallelogram.
(b)
Area square units
(c) not a rectangleAB\
� AC\
� 5 � 8 � 3 � 16 � 0 ⇒
� �AB\
� AC\
� � ���10�2 � 142 � ��6�2 � 2�83
AB\
� AC\
� � i15
j24
k31� � ��10, 14, �6�
ABCD
CD\
� �1, 2, 3�
AB\
� �1, 2, 3�
41. (a)is parallel to
is parallel to
(c)
� 0 ⇒ not a rectangle
AB\
� AD\
� �1, 2, �2� � ��3, 4, 4�
BC\
� ��3, 4, 4�.AD\
� ��3, 4, 4�
DC\
� �0 � ��1�, 5 � 3, 6 � 8� � �1, 2, �2�.
AB\
� �3 � 2, 1 � ��1�, 2 � 4� � �1, 2, �2� (b)
� �360 � 6�10 square units
� �162 � 22 � 102
Area � �AB\
� AD\
�
AB\
� AD\
� � i1
�3
j2
4
k�2
4� � �16, 2, 10�
43.
Area �12�u � v � �
12�81 � 36 �
32�13
u � v � � i1
�3
j20
k30� � �0, �9, 6�
u � �1, 2, 3�, v � ��3, 0, 0� 44.
square units � 12�396 � 3�11
Area �12�u � v� �
12���6�2 � 62 � 182
u � v � � i1
�3
j4
6
k�1
�3� � ��6, 6, 18�
v � ��2 � 1, 2 � ��4�, 0 � 3� � ��3, 6, �3�
u � �2 � 1, 0 � ��4�, 2 � 3� � �1, 4, �1�
©H
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.
910 Chapter 10 Analytic Geometry in Three Dimensions
45.
� 12�4290 square units
Area �12 �u � v � �
12���40�2 � 492 � 172
u � v � � i�4
1
j�5
�3
k5
11� � ��40, 49, 17�
v � �3 � 2, 0 � 3, 6 � ��5�� � �1, �3, 11�
u � ��2 � 2, �2 � 3, 0 � ��5�� � ��4, �5, 5� 46.
sq. units � 12�1280 � 8�5
Area �12�u � v� �
12���32�2 � 162
u � v � � i�4
�2
j�8
�4
k0
4� � ��32, 16, 0�
v � �0 � 2, 0 � 4, 4 � 0� � ��2, �4, 4�
u � ��2 � 2, �4 � 4, 0 � 0� � ��4, �8, 0�
47.
� 2�16� � 3�16� � 3�0� � �16
u � �v � w� � �2
4
0
3
4
0
3
0
4� 48. u � �v � w� � �200 0
3
0
1
0
1� � 6
49. u � �v � w� � �214
3�1
3
101� � 2��1� � 3�1� � 1�7� � 2
50. u � �v � w� � �120 40
�3
�746� � 1�0 � 12� � 4�12 � 0� � 7��6� � 6
51.
Volume � �u � �v � w�� � 2 cubic units
u � �v � w� � �1
0
1
1
1
0
0
1
1� � 1 � 1 � 2
53.
Volume � �u � �v � w�� � 12 cubic units
u � �v � w� � �003
200
2�2
2� � 0 � 2�6� � 2�0� � �12
52.
Volume cubic units� �u � �v � w�� � ��9� � 9
u � �v � w� � �103 1
3
0
3
3
3� � 1�9� � 1��9� � 3��9� � �9
54.
Volume cubic units� �u � �v � w�� � 16
u � �v � w� � � 1�1
2
220
�121� � 1�2� � 2��1 � 4� � 1(0 � 4� � 16
©H
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Section 10.3 The Cross Product of Two Vectors 911
55.
Volume � ��84� � 84 cubic units
u � �v � w� � �4
0
0
0
�2
5
0
3
3� � 4��21� � �84
w � �0, 5, 3�v � �0, �2, 3�,u � �4, 0, 0�,
57.
(a)
(b)
T�p� � �V � F� �p
2 cos 40�
� p2
cos 40��iV � F � � i0
0
j �
12 cos 40�
0
k�
12 sin 40�
�p
�
56.
Volume cubic units� 3
u � �v � w� � �110 101
021� � 1��2� � 1�1� � �3
AB\
� �1, 1, 0�, AC\
� �1, 0. 2�, AD\
� �0, 1, 1�
15 20 25 30 35 40 45
5.75 7.66 9.58 11.49 13.41 15.32 17.24T
p
58.
ft-lb �PQ\
� F� � 160�3
PQ\
� F � � i0
0
j0
�1000�3
k0.16
�1000 � � 160�3 i
PQ\
� 0.16k
y
x
PQ
0.16 ft
F°60
z
59. True. The cross product is defined for vectors inthree-dimensional space.
60. False. u � v � ��v � u�
61. If and are orthogonal, then and hence, �u � v� � �u� �v� sin � � �u� �v�.sin � � 1vu
62.
� �u � w�v � �u � v�w
� �u1w1 � u2w2 � u3w3�v � �u1v1 � u2v2 � u3v3�w
� �u1v1 � u2v2 � u3v3��w1i � w2 j � w3k� � �u1w1 � u2w2 � u3w3��v1i � v2 j � v3k�
� �u1w1v3 � u2w2v3 � u1v1w3 � u2v2w3 k� ��u1w1v2 � u3w3v2 � u1v1w2 � u3v3w2 j
� �u2w2v1 � u3w3v1 � u2v2w1 � u3v3w1 i
� �u1�w1v3 � v1w3 � � u2�v2w3 � w2v3 � k� �u1�v1w2 � w1v2 � � u3�v2w3 � w2v3 � j
� �u2�v1w2 � w1v2 � � u3�w1v3 � v1w3 �i
� � iu1
v2w3 � w2v3
ju2
w1v3 � v1w3
ku3
v1w2 � w1v2�u � �v � w� � u � � i
v1
w1
jv2
w2
k v3
w3� � u � ��v2w3 � w2v3�i � �v1w3 � w1v3�j � �v1w2 � w1v2�k
©H
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.
912 Chapter 10 Analytic Geometry in Three Dimensions
63.
Hence,
� � u1
v1
w1
u2
v2
w2
u3
v3
w3�. u � �v � w� � u1�v2w3 � w2v3� � u2�v1w3 � w1v3� � u3�v1w2 � v2w1�
v � w � � i v1
w1
jv2
w2
k v3
w3� � �v2w3 � w2v3�i � �v1w3 � w1v3�j � �v1w2 � v2w1�k
64.
Area of triangle formed by the unit vectors u and v is
The area is also given by
Notice that is negative.
Thus, sin�� � �� � sin � cos � � cos � sin �.
cos � sin � � sin � cos �
12�u � v� �
12�cos � sin � � sin � cos ��
12�base��height� �
12�1� sin�� � ��.
u
v
α
α
β
β−
x
y
u � v � � icos � cos �
jsin �sin �
k00� � �cos � sin � � sin � cos ��k
65. cos 480� � cos 120� � �12 67. sin 690� � sin 330� � �
12
69. sin 19
6� sin�7
6 � � �12
71. tan 15
4� tan
7
4� �1
66. tan 300� � ��3
68. cos 930� � cos 210� � ��32
70. cos 17
6� cos
5
6� �
�32
72. tan 10
3� tan
4
3� �3
Section 10.4 Lines and Planes in Space
■ The parametric equations of the line in space parallel to the vector and passing through the pointare
■ The standard equation of the plane in space containing the point and having normal vectoris
■ You should be able to find the angle between two planes by calculating the angle between theirnormal vectors.
■ You should be able to sketch a plane in space.
■ The distance between a point Q and a plane having normal n is
where P is a point in the plane.
D � �projn PQ\
� ��PQ
\
� n��n�
a�x � x1� � b� y � y1� � c�z � z1� � 0.
�a, b, c��x1, y1, z1�
z � z1 � ct.y � y1 � bt,x � x1 � at,
�x1, y2, z3��a, b, c
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Section 10.4 Lines and Planes in Space 913
Vocabulary Check
1. direction, 2. parametric equations 3. symmetric equations
4. normal 5. a�x � x1� � b�y � y1� � c�z � z1� � 0
PQ\
t
1.
(a) Parametric equations:
(b) Symmetric equations:x1
�y2
�z3
x � t, y � 2t, z � 3t
z � z1 � ct � 0 � 3t
y � y1 � bt � 0 � 2t
x � x1 � at � 0 � t 2.
(a) Parametric equations:
(b) Symmetric equations:x � 3
3�
y � 5�7
�z � 1�10
x � 3 � 3t, y � �5 � 7t, z � 1 � 10t
z � z1 � ct � 1 � 10t
y � y1 � bt � �5 � 7t
x � x1 � at � 3 � 3t
3.
(a) Parametric equations:
Equivalently:
(b) Symmetric equations:x � 4
3�
y � 1
8�
z
�6
x � �4 � 3t, y � 1 � 8t, z � �6t
z � �ty � 1 �4
3t,x � �4 �
1
2t,
z � z1 � ct � 0 � ty � y1 � bt � 1 �4
3t,x � x1 � at � �4 �
1
2t,
4.
(a) Parametric equations:
(b)
Not possible
x � 5
4�
z � 10
3, y � 0
z � 10 � 3ty � 0,x � 5 � 4t,
z � z1 � ct � 10 � 3t
y � y1 � bt � 0 � 0t
x � x1 � at � 5 � 4t 5.
(a) Parametric equations:
(b) Symmetric equations:x � 2
2�
y � 3
�3� z � 5
x � 2 � 2t, y � �3 � 3t, z � 5 � t
z � z1 � ct � 5 � t
y � y1 � bt � �3 � 3t,
x � x1 � at � 2 � 2t,
6.
(a)
(b) Symmetric equations:x � 1
3�
y�2
�z � 1
1
x � 1 � 3t, y � �2t, z � 1 � t
v � �3, �2, 1 7.
Point:
(a)
(b)x � 2�1
�y4
�z � 2�5
x � 2 � t, y � 4t, z � 2 � 5t
�2, 0, 2�
v � �1 � 2, 4 � 0, �3 � 2 � ��1, 4, �5
8.
Point:
(a) Parametric equations:
(b) Symmetric equations:x � 2
8�
y � 35
�z
12
x � 2 � 8t, y � 3 � 5t, z � 12t
�2, 3, 0�
v � �8, 5, 12 9.
Point:
(a)
(b)x � 3
4�
y � 8�10
�z � 15
1
x � �3 � 4t, y � 8 � 10t, z � 15 � t
��3, 8, 15�
v � �1 � ��3�, �2 � 8, 16 � 15 � �4, �10, 1
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914 Chapter 10 Analytic Geometry in Three Dimensions
10.
Point:
(a)
(b)x � 2�1
�y � 3�8
�z � 1
4
x � 2 � t, y � 3 � 8t, z � �1 � 4t
�2, 3, �1�
v � �1 � 2, �5 � 3, 3 � 1 � ��1, �8, 4 11.
Point:
(a)
(b)
Not possible
x � 3�4
�z � 2
3, y � 1
x � 3 � 4t, y � 1, z � 2 � 3t
�3, 1, 2�
v � ��1 � 3, 1 � 1, 5 � 2 � ��4, 0, 3
12.
Point:
(a)
(b)
Not possible
y � 12
�z � 5�8
, x � 2
x � 2, y � �1 � 2t, z � 5 � 8t
�2, �1, 5�
v � �2 � 2, 1 � 1, �3 � 5 � �0, 2, �8 13.
or
Point:
(a)
(b)x �
12
3�
y � 2�5
�z �
12
�1
x � �12
� 3t, y � 2 � 5t, z �12
� t
��12
, 2, 12�
�3, �5, �1
v � 1 �12
, �12
� 2, 0 �12� � 3
2, �
52
, �12�
14. or
Point:
(a) Parametric equations:
(b) Symmetric equations:x � 3
9�
y � 5�13
�z � 4�12
x � 3 � 9t, y � �5 � 13t, z � �4 � 12t
�3, �5, �4�
�9, �13, �12v � 3 � ��32�, �5 �
32
, �4 � 2� � 92
, �132
, �6�,
15.
y
x
(0, 2, 1)
z
y
x
3
2
1
1
−1
−2−1−2
−3
12
3
23
z 16.
(5, 1, 5)
y
x
6
4
2
−4−2−4
−6
46
z
17.
x � 2 � 0
1�x � 2� � 0�y � 1� � 0�z � 2� � 0
a�x � x1� � b�y � y1� � c�z � z1� � 0
19.
�2x � y � 2z � 10 � 0
�2�x � 5� � 1�y � 6� � 2�z � 3� � 0
21.
�x � 2y � z � 2 � 0
�1�x � 2� � 2�y � 0� � 1�z � 0� � 0
n � ��1, �2, 1 ⇒
18.
z � 3 � 0
0�x � 1� � 0�y � 0� � 1�z � 3� � 0
a�x � x0� � b�y � y0� � c�z � z0� � 0
20.
�3y � 5z � 0
0�x � 0� � 3�y � 0� � 5�z � 0� � 0
22.
�x � y � 2z � 12 � 0
�1�x � 0� � 1�y � 0� � 2�z � 6� � 0
n � ��1, 1, �2
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Section 10.4 Lines and Planes in Space 915
23.
3x � 9y � 7z � 0
�3x � 9y � 7z � 0
�3�x � 0� � 9�y � 0� � 7�z � 0� � 0
n � u � v � � i1
�2
j23
k33� � ��3, �9, 7
v � ��2 � 0, 3 � 0, 3 � 0 � ��2, 3, 3
u � �1 � 0, 2 � 0, 3 � 0 � �1, 2, 3 24.
�x � y � 4z � 7 � 0
Plane: �1�x � 4� � 1�y � 1� � 4�z � 3� � 0
n � ��1, 1, 4
u � v � � i2
�3
j�6
�3
k2
0� � �6, �6, �24
v � ��3, �3, 0u � �2, �6, 2,
25.
6x � 2y � z � 8 � 0
18x � 6y � 3z � 24 � 0
18�x � 2� � 6�y � 3� � 3�z � 2� � 0
n � u � v � � i1
�1
j1
�4
k42� � �18, �6, �3
v � �1 � 2, �1 � 3, 0 � 2 � ��1, �4, 2
u � �3 � 2, 4 � 3, 2 � 2 � �1, 1, 4 26.
Plane:
�2x � 11y � 4z � 5 � 0
�2�x � 1� � 11�y � 1� � 4�z � 2� � 0
n � ��2, 11, 4
u � v � � i41
j02
k2
�5� � ��4, 22, 8
u � �4, 0, 2, v � �1, 2, �5
27.
y � 5 � 0
n � j: 0�x � 2� � 1�y � 5� � 0�z � 3� � 0 28. normal to -plane
x � 1 � 0
1�x � 1� � 0� y � 2� � 0�z � 3� � 0
yzn � �1, 0, 0,
29. andare parallel to the plane.
y � z � 2 � 0
4y � 4z � 8 � 0
0�x � 0� � 4�y � 2� � 4�z � 4� � 0
n � � i11
j40
k40� � �0, 4, �4
�1, 0, 0�0 � ��1�, 2 � ��2�, 4 � 0 � �1, 4, 4
32. andare parallel to the plane.
x � 2y � 4z � 5 � 0
�3x � 6y � 12z � 15 � 0
�3�x � 1� � 6�y � 2� � 12�z � 0� � 0
n � � i22
j3
�3
k�2
1� � ��3, �6, �12
�2, �3, 1�1 � ��1�, 2 � ��1�, 0 � 2 � �2, 3, �2
30. and are parallel to the plane.
5x � 3z � 17 � 0
5�x � 1� � 0�y � 2� � 3�z � 4� � 0
n � � i30
j21
k�5
0� � �5, 0, 3
�0, 1, 0�4 � 1, 0 � ��2�, �1 � 4 � �3, 2, �5
31. andare parallel to plane.
�7x � y � 11z � 5 � 0
�7�x � 2� � 1�y � 2� � 11�z � 1� � 0
n � � i�3
2
j�1�3
k�2
1� � ��7, �1, 11
�2, �3, 1��1 � 2, 1 � 2, �1 � 1 � ��3, �1, �2
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916 Chapter 10 Analytic Geometry in Three Dimensions
33. and
z � 4 � t
y � 3
x � 2
P � �2, 3, 4�v � �0, 0, 1 34. and
z � 2
y � 5 � t
x � �4
P � ��4, 5, 2�v � �0, 1, 0 35. and
z � 4 � t
y � 3 � 2t
x � 2 � 3t
P � �2, 3, 4�v � �3, 2, �1
36. and
z � 2 � t
y � 5 � 2t
x � �4 � t
P � ��4, 5, 2�v � ��1, 2, 1 37. and
z � �4 � 3t
y � �3 � t
x � 5 � 2t
P � �5, �3, �4�v � �2, �1, 3 38. and
z � �3
y � 4 � t
x � �1 � 5t
P � ��1, 4, �3�v � �5, �1, 0
39. and
z � 2 � t
y � 1 � t
x � 2 � t
P � �2, 1, 2�v � ��1, 1, 1 40. and
z � 8
y � 2t
x � �6 � 2t
P � ��6, 0, 8�v � ��2, 2, 0
41.
orthogonaln1 � n2 � 5 � 12 � 7 � 0;
n1 � �5, �3, 1, n2 � �1, 4, 7
43.
orthogonaln1 � n2 � 8 � 8 � 0;
n2 � �4, 1, 8n1 � �2, 0, �1,
42.
parallel planes3n1 � �9, 3, �12 � �n2 ⇒ n2 � ��9, �3, 12n1 � �3, 1, �4,
44.
paralleln2 � �5, �25, �5 � 5n1 ⇒
n1 � �1, �5, �1
45. (a) normal vectors to planes
(b) Equation 1
Equation 2
times Equation 2 added to Equation 1 gives
Substituting back into Equation 2,
Letting we obtain x � 2 � t, y � 8t, z � 7t.t � z�7,
x � 2 � y � z � 2 �87
z � z � 2 �17
z.
y �87
z.
�7y � 8z � 0
��3�
x � y � z � 2
3x � 4y � 5z � 6
60.67�⇒cos ��n1 � n2��n1� �n2�
���6�
�50�3�
6�150
n1 � �3, �4, 5, n2 � �1, 1, �1;
46. (a)
—CONTINUED—
⇒ 66.93�cos ��n1 � n2��n1� �n2�
��7�
�11�29�
7
�319
n1 � �1, �3, 1, n2 � �2, 0, 5
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Section 10.4 Lines and Planes in Space 917
47. (a) normal vectors to planes
(b) Equation 1
Equation 2
times Equation 1 added to Equation 2 gives
Substituting back into Equation 1,
Letting Equivalently, let and x � 6t � 1.y � t, z � 7t � 1x �6t � 1
7, y �
t � 17
.z � t,
x � z � y � z �z � 1
7�
6z7
�17
�17
�6z � 1�.
y �z � 1
7.
�7y � z � 1
��2�
2x � 5y � z � 1
x � y � z � 0
77.83�⇒cos ��n1 � n2��n1� �n2�
���2�
�3�30�
2�90
n1 � �1, 1, �1, n2 � �2, �5, �1;
48. The planes are parallel because is a multiple of The planes do not intersect.n2 � ��3, �6, 3.n1 � �2, 4, �2
49.
y
x
34
4
56
2
−2
56
23
(0, 0, 2)
(0, 3, 0)
(6, 0, 0)
z
x � 2y � 3z � 6 51.
y
x
32
4
4
6
3
−1−2
564
32
56
(0, 2, 0)(4, 0, 0)
z
−1−2
x � 2y � 4
53.
x y6
−1−2 −2−1
−6−7
54
3
65
4
z
(0, 0, −6)
(0, 3, 0)(2, 0, 0)
3x � 2y � z � 6
50.
y
x
3456
23
−5−2
23
45
6
−4
(0, 4, 0)−(0, 0, 1)
(2, 0, 0)
z
2x � y � 4z � 4
52.
y
x
32
6
32
−1 −1−2
−2
6
23
45
6
(0, 5, 0)
(0, 0, 5)
z
y � z � 5 54.
xy
2
2
−2
−2
−4
−6
6
z
(6, 0, 0)(0, 0, −2)
x � 3z � 6
46. —CONTINUED—
(b)
Then
Let Parametric equations:
Or equivalently, let and you obtain x � �5t �32
, y � �t �16
, z � 2t.z � 2t
z � ty � �12
t �16
,x � �52
t �32
,z � t.
y � �12
z �16
. ⇒ 3y � x � z � 2 �12
��5z � 3� � z � 2 � �32
z �12
2x � 5z � 3 � 0 ⇒ x �12
��5z � 3�
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918 Chapter 10 Analytic Geometry in Three Dimensions
55.
on plane,
D ����1, 0, 0 � �8, �4, 1�
�64 � 16 � 1�
��8��81
�89
n � �8, �4, 1, PQ\
� ��1, 0, 0
Q � �0, 0, 0�,P � �1, 0, 0�
D ��PQ
\
� n��n�
56. on plane,
D � �PQ\
� n��n�
� ��1��6
�1
�6�
�6
6
PQ\
� ��1, 2, 1
Q � �3, 2, 1�, n � �1, �1, 2P � �4, 0, 0�
57.
on plane,
D ���2, �2, �2 � �2, �1, 1�
�6�
4�6
�2�6
3
PQ\
� �2, �2, �2n � �2, �1, 1,
Q � �4, �2, �2�,P � �2, 0, 0�
D ��PQ
\
� n��n�
58. on plane,
D ��PQ
\
� n��n�
� ��3��14
�3
�14�
3�1414
n � �2, 3, 1PQ\
� ��7, 2, 5 ,
Q � ��1, 2, 5�,P � �6, 0, 0�
59. The normal vector to plane containing and is obtained as follows.
The normal vector to the plane containing and is obtained as follows.
The angle between two adjacent sides is given by
88.45�.⇒cos ��n1 � n2��n1� �n2�
���1�
�37�37�
137
n2 � ��6, 0, 1
u1 � u2 � � i20
j2
10
k120� � ��120, 0, 20
u1 � �2, 2, 12, u2 � �0, 10, 0
�0, 10, 0��0, 0, 0�, �2, 2, 12�
n1 � �0, 6, �1
v1 � v2 � � i2
10
j20
k120� � �0, 120, �20
v1 � �2, 2, 12, v2 � �10, 0, 0
�10, 0, 0��0, 0, 0�, �2, 2, 12�
60. The plane containing has normal vector
or
The plane containing and has normal vector
or
The angle between two adjacent sides is given by
89.12�. ⇒ cos ��n1 � n2��n1� �n2�
�1
�65�65�
165
n2 � ��8, 0, 1.�0, �6, 0 � �1, 1, 8 � � i01
j�6
1
k08� � ��48, 0, 6,
R�7, 7, 8�Q�6, 6, 0�,P�6, 0, 0�,
n1 � �0, 8, 1.�6, 0, 0 � ��1, �1, 8 � � i6
�1
j0
�1
k08� � �0, �48, �6
S�0, 0, 0�, T��1, �1, 8�P�6, 0, 0�,©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Review Exercises for Chapter 10 919
65. x2 � y2 � 102 � 100
67. , , x2 � y2 � 3xr2 � 3r cos �r � 3 cos �
69.
r � 7
r2 � 49
71.
r � 5 csc �
r sin � � 5
y � 5
64. (a) Sphere:
(b) Two planes parallel to given plane. Let be a point on one of these planes, and pick on the given plane. By the distance formula,
(Two planes parallel to given plane)4x � 3y � z � 10 ± 2�26
±2�26 � 4x � 3y � z � 10
2 � �PQ\
� n��n�
� ��x, y, z � 10� � �4, �3, 1���26
P � �0, 0, 10�Q � �x, y, z�
�x � 4�2 � �y � 1�2 � �z � 1�2 � 4
66. (line)y � �x⇒tan � � �1 �yx
⇒� �3�
4
68.
⇒ 2�x2 � y2 � x � 1 ⇒ 4�x2 � y2� � x2 � 2x � 1 ⇒ 3x2 � 4y2 � 2x � 1
r �1
2 � cos � ⇒ 2r � r cos � � 1 ⇒ 2�x2 � y2 � x � 1
70.
r � 4 cos �⇒ r � 4 cos � � 0
r2 � 4r cos � � 0
x2 � y2 � 4x � 0
72.
r �1
sin � � 2 cos �
r�2 cos � � sin �� � �1
2r cos � � r sin � � �1
2x � y � 1 � 0
Review Exercises for Chapter 10
1. (a) and (b)
y
x
12
34
1
2
3
−2
−3
−4
−5
1 2 3−2
−3
−4−5
(5, −1, 2)
(−3, 3, 0)
z 2.
y
x
12
34
1
2
−2
−3
3
4
5
31 2−2
−3−4
−3−4
z
(0, 0, 5)
(2, 4, −3)
61. False. They might be skew lines, such as:
(x-axis)
and L2: x � 0, y � t, z � 1
L1: x � t, y � 0, z � 0
62. True
63. The lines are parallel: �32�10, �18, 20� � ��15, 27, �30�
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920 Chapter 10 Analytic Geometry in Three Dimensions
3. ��5, 4, 0� 4. -axis
�0, �7, 0�
⇒ x � z � 0y
5.
� �41
� �1 � 4 � 36
d � ��5 � 4�2 � �2 � 0�2 � �1 � 7�2 6.
� �61
� �9 � 36 � 16
d � ��2 � ��1��2 � �3 � ��3��2 � ��4 � 0�2
7.
d12 � d2
2 � 38 � 29 � 67 � d32
d3 � ��0 � 3�2 � �5 � ��2��2 � ��3 � 0�2 � �9 � 49 � 9 � �67
d2 � ��0 � 0�2 � �5 � 3�2 � ��3 � 2�2 � �4 � 25 � �29
d1 � ��3 � 0�2 � ��2 � 3�2 � �0 � 2�2 � �9 � 25 � 4 � �38
8.
d1 2 � d2
2 � d3 2 � 42
d3 � ��4 � 0�2 � �5 � 0�2 � �5 � 4�2 � �16 � 25 � 1 � �42
d2 � ��4 � 4�2 � �5 � 3�2 � �5 � 2�2 � �4 � 9 � �13
d1 � ��4 � 0�2 � �3 � 0�2 � �2 � 4�2 � �16 � 9 � 4 � �29
9. Midpoint: ��2 � 22
, 3 � 5
2,
2 � ��2�2 � �0, �1, 0� 10. Midpoint: �7 � 1
2,
1 � 12
, �4 � 2
2 � �4, 0, �1�
11. Midpoint: �10 � 82
, 6 � 2
2,
�12 � 62 � �1, 2, �9� 12. Midpoint: ��5 � 7
2, �3 � 9
2,
1 � 52 � ��6, �6, �2�
13. �x � 2�2 � �y � 3�2 � �z � 5�2 � 1 14. �x � 3�2 � �y � 2�2 � �z � 4�2 � 16
15. Radius: 6
�x � 1�2 � �y � 5�2 � �z � 2�2 � 36
16. Radius
x2 � �y � 4�2 � �z � 1�2 �2254
�152
17.
Center:
Radius: 3
�2, 3, 0�
�x � 2�2 � �y � 3�2 � z2 � 9
�x2 � 4x � 4� � �y2 � 6y � 9� � z2 � �4 � 4 � 9
18.
Center:
Radius: 2
�5, �3, 2�
�x � 5�2 � �y � 3�2 � �z � 2�2 � 4
�x2 � 10x � 25� � �y2 � 6y � 9� � �z2 � 4z � 4� � �34 � 25 � 9 � 4
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Review Exercises for Chapter 10 921
19. (a) xz-trace circle
2
−2− 42
4
4 46x
y
(0, 3, 0)
x z2 2+ = 7
z
x2 � z2 � 7,�y � 0�: (b) yz-trace circle
2
−2 −2
2
4
4 42
6x
y
( 3) + = 16y z− 2 2
(0, 3, 0)
z
�y � 3�2 � z2 � 16,�x � 0�:
20. (a) -trace circle (b) yz-trace
circle
2
4
6
x
y
( 1) + = 5y z− 2 2
( 2, 1, 0)−
z
�y � 1�2 � z2 � 5,
2
4
64x
y
( + 2) + ( 1) = 9x y2 2−
( 2, 1, 0)−
z
4 � �y � 1�2 � z2 � 9�x � 0�:�x � 2�2 � �y � 1�2 � 9,�z � 0�:xy
21. (a)
(b)
(c) Unit vector:�3333
�1, 4, �4�
�v� � �12 � 42 � ��4�2 � �33
v � �3 � 2, 3 � ��1�, 0 � 4� � �1, 4, �4� 22. (a)
(b)
(c) Unit vector:�3535
��5, 3, 1�
�v� � ���5�2 � 32 � 12 � �35
v � ��3 � 2, 2 � ��1�, 3 � 2� � ��5, 3, 1�
23. (a)
(b)
(c) Unit vector:�185185
��10, 6, 7�
�v� � ���10�2 � 62 � 72 � �185
v � ��3 � 7, 2 � ��4�, 10 � 3� � ��10, 6, 7� 24. (a)
(b)
(c) Unit vector:�195195
�5, �11, 7�
�v� � �52 � ��11�2 � 72 � �195
v � �5 � 0, �8 � 3, 6 � ��1�� � �5, �11, 7�
25. u � v � �1�0� � 4��6� � 3�5� � �9 26. u � v � 8�2� � 4�5� � 2�2� � 0
27. u � v � 2�1� � 1�0� � 1��1� � 1 28. u � v � 2�1� � 1��3� � 2�2� � �5
29.
The vectors are orthogonal.
� 90�
cos � �u � v
�u� �v��
12 � 2 � 10
�42�17� 0 30.
The vectors are parallel.
� 180�
cos � �u � v
�u� �v��
�20 � 5 � 45
�350�14�
�70
70� �1
31. Since the angle is 90�.u � v � 0, 32.
�15
�11�45 ⇒ � 47.61�
cos � �u � v
�u� �v��
12 � 5 � 2
�11�45
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922 Chapter 10 Analytic Geometry in Three Dimensions
33.
Orthogonal
u � v � 7��1� � ��2��4� � 3�5� � 0 34.
Parallel
�4u � �4��4, 3, �6� � �16, �12, 24� � v
35. Since thevectors are parallel.
�23�39, �12, 21� � ��26, 8, �14�, 36.
Orthogonal
� �16 � 20 � 4 � 0
u � v � �8, 5, �8� � ��2, 4, 12�
37. First two points:
Last two points:
Since the points are not collinear.u � cv,
v � �0, �2, 6�
u � ��3, 4, 1� 38. First two points:
Last two points:
Since, the threepoints are collinear.
�2, �10, �8� � �2��1, 5, 4�,
�2, �10, �8�
��1, 5, 4�
40. First two points:
Last two points:
Since the three pointsare not collinear.
�3, �1, �2� � c�3, 11, �2�,
�3, 11, �2�
�3, �1, �2�39. First two points:
First and third points:
Since the threepoints are collinear.
�4, �2, �10� � 2�2, �1, �5�,
�2, �1, �5�
�4, �2, �10�
41. Let a, b, and c be the three force vectors determined by and
Must have Thus,
From the first equation, From the second equation,
From the third equation, Thus,
and
Finally, �a� � �2� 6�38�
75�384 �
225�22
159.10.
�b� � �c� �75�38
4 115.58.
16�38
�b� � 300⇒6�38
�b� � 300 �10�38
�b�
1�2
�a� � 300 �10�38
�b�.
1�2
�a� �6
�38 �b�.�b� � �c�.
1�2
�a� �5
�38 �b� �
5�38
�c� � 300.
1�2
�a� �3
�38 �b� �
3�38
�c� � 0
�2�38
�b� �2
�38 �c� � 0
a � b � c � 300k.
c � �c��4, �6, 10��152
� �c�� 2�38
, �3�38
, 5
�38�
b � �b���4, �6, 10�
�152� �b� � �2
�38,
�3�38
, 5
�38�
a � �a��0, 10, 10�
10�2� �a��0,
1�2
, 1�2�
C�4, �6, 10�.B��4, �6, 10�,A�0, 10, 10�,
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Review Exercises for Chapter 10 923
42. Let a, b, c be the three force vectors determined by and
We must have Thus,
Solving this system, Thus, the tensions are and 77.1 pounds.106.1, 77.1�a� 106.1, �b� � �c� � 77.1.
1�2
�a� �5
�38�b� �
5�38
�c� � 200
1�2
�a� �3
�38�b� �
3�38
�c� � 0
�2�38
�b� �2
�38�c� � 0
a � b � c � 200k.
c � �c��4, �6, 10��152
� �c�� 2�38
, �3�38
, 5
�38�
b � �b���4, �6, 10�
�152� �b�� �2
�38,
�3�38
, 5
�38�
a � �a��0, 10, 10�
10�2� �a��0,
1�2
, 1�2�
C�4, �6, 10�.B��4, �6, 10�,A�0, 10, 10�,
43. u v � � i�2
1
j8
1
k2
�1� � ��10, 0, �10� 44. u v � � i10
5
j15
�3
k5
0� � �15, 25, �105�
45.
Unit vector:1
�7602 ��71, �44, 25�
�u v � � �7602
u v � � i�310
j2
�15
k�5
2� � ��71, �44, 25�
46. unit vector: j � �0, 1, 0� ⇒ u v � � i01
j00
k4
12� � 4 j
47. First two points:
Last two points:
First and third points:
� 2�43 square units
� �172
� �36 � 36 � 100
Area � ���6, �6, 10��
� i3
�2
j22
k30� � ��6, �6, 10�
��2, 2, 0�
�3, 2, 3�
�3, 2, 3� 48.
Opposite sides parallel and equal length
Adjacent sides:
Area � �u w� � �4 � 4 � 2�2 square units
u w � � i10
j02
k10� � ��2, 0, 2�
u � �1, 0, 1�, w � �0, 2, 0�
u � �1, 0, 1�, v � �1, 0, 1�,
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924 Chapter 10 Analytic Geometry in Three Dimensions
49. The parallelogram is determined by the threevectors with initial point
Volume � ��75� � 75 cubic units
u � �v w� � �320
005
051� � �75
u � �3, 0, 0�, v � �2, 0, 5�, w � �0, 5, 1�
�0, 0, 0�.50.
Volume cubic units� �u � �v w�� � 48
u � �v w� �
2�00 040
006� � 48
u � �2, 0. 0�, v � �0, 4, 0�, w � �0, 0, 6�
53. point:
(a) Parametric equations:
(b) Symmetric equations:x � 1
4�
y � 33
�z � 5�6
x � �1 � 4t, y � 3 � 3t, z � 5 � 6t
��1, 3, 5�v � �3 � 1, 6 � 3, �1 � 5� � �4, 3, �6�,
51.
Point:
(a)
(b)x � 3
6�
y11
�z � 2
4
z � 2 � 4ty � 11t,x � 3 � 6t,
�3, 0, 2�
v � �9 � 3, 11 � 0, 6 � 2� � �6, 11, 4� 52.
Point:
(a)
(b)x � 1
9�
y � 46
�z � 3
2
z � 3 � 2ty � 4 � 6t,x � �1 � 9t,
��1, 4, 3�
v � �9, 6, 2�
54. (a)
(b)x5
�y � 10
20�
z � 3�3
z � 3 � 3ty � �10 � 20t,x � 5t,
v � �5, 20, �3� 55. Use point:
(a) Parametric equations:
(b) Symmetric equations:x
�4�
y5
�z2
x � �4t, y � 5t, z � 2t
�0, 0, 0�.2v � ��4, 5, 2�,
56. (a)
(b) or
x � 3 � y � 2 � z � 1
x � 31
�y � 2
1�
z � 11
x � 3 � t, y � 2 � t, z � 1 � t
v � �1, 1, 1� 57.
2x � 12y � 5z � 0
2�x � 0� � 12�y � 0� � 5�z � 0� � 0
a�x � x0� � b�y � y0� � c�z � z0� � 0
n � �2, 12, �5�
u v � � i52
j03
k28� � ��6, �36, 15�
u � �5, 0, 2�, v � �2, 3, 8�
58.
�2y � 5z � 14 � 0
�2�y � 3� � 5�z � 4� � 0
Plane: 0�x � 1� � 16�y � 3� � 40�z � 4� � 0
n � u v � � i5
3
j�5
5
k�2
2� � �0, �16, 40�
v � �3, 5, 2�u � �5, �5, �2�, 59. normal vector
z � 2 � 0
Plane: 0�x � 5� � 0�y � 3 � � 1 �z � 2 � � 0
n � k,
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Review Exercises for Chapter 10 925
60. point:
x � y � 2z � 12 � 0
�x � y � 2z � 12 � 0
�1�x � 0� � 1�y � 0� � 2�z � 6� � 0
�0, 0, 6�n � ��1, 1, �2�,
61.
x
y
(0, 0, 2)
(2, 0, 0)
(0, 3, 0)−
11
2
3
1
−2
z
3x � 2y � 3z � 6 62.
y
x
24
−4
2
4
−2
−42
4
(1, 0, 0)
(0, 0, 1)−
(0, 5, 0)−
z
5x � y � 5z � 5
63.
x
y
(3, 0, 0)
(0, 0, 2)−1 2
1
−2 −1
3
3
4
1
−1
−2
2
z
2x � 3z � 6 64.
x
y
(0, 0, −4)
(0, 3, 0)1 21
−1
−2
43
2
−2
−3
−1
1
z
4y � 3z � 12
65. in plane,
D ��PQ
\
� n��n�
���2��440
�1
�110�
�110
110 0.0953
PQ\
� �2, 3, 9�Q � �2, 3, 10�,n � �2, �20, 6�, P � �0, 0, 1�
66.
in plane,
D ����1, 2, 3� � �2, �1, 1��
�6�
1�6
��66
PQ\
� ��1, 2, 3�, n � �2, �1, 1�Q � �1, 2, 3�, P � �2, 0, 0�
D ��PQ
\
� n��n�
67. in plane,
D ��PQ
\
� n��n�
���2�
�1 � 100 � 9�
2�110
�2�110
110�
�11055
0.191
Q � �0, 0, 0�, PQ\
� ��2, 0, 0�n � �1, �10, 3�, P � �2, 0, 0�
68.
in plane,
D ���0, 0, �12� � �2, 3, 1��
�14�
12�14
�6�14
7
PQ\
� �0, 0, �12�, n � �2, 3, 1�Q � �0, 0, 0�, P � �0, 0, 12�
D ��PQ
\
� n��n�
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926 Chapter 10 Analytic Geometry in Three Dimensions
69. False. a b � ��b a� 70. True. See page 761.
71.
� �u�2
� 14
� 9 � 4 � 1
u � u � �3, �2, 1� � �3, �2, 1� 72.
Thus, u v � ��v u�.
v u � � i23
j�4�2
k�3
1� � ��10, �11, 8�
u v � � i32
j�2�4
k1
�3� � �10, 11, �8�
73.
u � v � u � w � 11 � ��5� � 6
u � �v � w� � �3, �2, 1� � �1, �2, �1� � 6
74.
(Exercise 72)
� u �v � w�
�u v� � �u w� � �10, 11, �8� � ��6, �7, 4� � �4, 4, �4�
u w � � i3
�1
j�2
2
k12� � ��6, �7, 4�
u v � �10, 11, �8�
u �v � w� � u �1, �2, �1� � � i31
j�2�2
k1
�1� � �4, 4, �4�
75. � �u2v3 � u3v2 �i � �u1v3 � u3v1�j � �u1v2 � u2v1�ku v � � i u1
v1
ju2
v2
k u3
v3�76. See table on page 759. 77. The magnitude will increase by a factor of 4.
78. Form vectors for two sides and complete their cross product.
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Practice Test for Chapter 10 927
Chapter 10 Practice Test
1. Find the lengths of the sides of the triangle with vertices and Show that the triangle is a right triangle.
�0, �2, �1�.�1, 2, �4�,�0, 0, 0�,
2. Find the standard form of the equation of a sphere having center and radius 5.�0, 4, 1�
3. Find the center and radius of the sphere x2 � y2 � z2 � 2x � 4z � 11 � 0.
4. Find the vector given and v � �4, 3, �6�.u � �1, 0, �1�u � 3v
5. Find the length of if v � �2, 4, �6�.12v
6. Find the dot product of and v � �1, 1, �2�.u � �2, 1, �3�
7. Determine whether and are orthogonal, parallel, or neither.v � ��3, �3, 3�u � �1, 1, �1�
8. Find the cross product of and What is v � u?v � �1, �1, 3�.u � ��1, 0, 2�
9. Use the triple scalar product to find the volume of the parallelepiped having adjacent edges and w � �1, 0, 4�.v � �0, �1, 1�,
u � �1, 1, 1�,
10. Find a set of parametric equations for the line through the points and �2, �3, 4�.�0, �3, 3�
11. Find an equation of the plane passing through and perpendicular to the vector n � �1, �1, 0�.�1, 2, 3�
12. Find an equation of the plane passing through the three points and C � �1, 2, 3�.B � �1, 1, 1�,A � �0, 0, 0�,
13. Determine whether the planes and are parallel, orthogonal, or neither.3x � 4y � z � 9x � y � z � 12
14. Find the distance between the point and the plane x � 2y � z � 6.�1, 1, 1�
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