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CHAPTER 10 Analytic 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 © Houghton Mifflin Company. All rights reserved.
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Page 1: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 2: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

888

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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Page 3: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 4: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 5: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 6: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 7: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 8: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 9: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 11: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 12: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 13: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 14: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 15: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 18: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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�

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Page 19: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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 �

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Page 20: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 21: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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.

Page 22: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 23: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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�

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Page 24: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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

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Page 25: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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

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Page 26: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 27: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 28: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 29: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 31: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 32: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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�,©

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Page 33: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 34: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 35: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 36: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 37: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 38: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 39: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 40: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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Page 41: az01001825.schoolwires.net...888 CHAPTER 10 Analytic Geometry in Three Dimensions Section 10.1 The Three-Dimensional Coordinate System Vocabulary Check 1. three-dimensional 2. xy-plane,

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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