C H A P T E R 1 1 Vectors and the Geometry of Space Section 11.1
Section 11.2
Section 11.3
Section 11.4
Section 11.5
Section 11.6
Section 11.7 Vectors in the Plane……………………………………………………………………..2
Space Coordinates and Vectors in Space ……………………………………..13
The Dot Product of Two Vectors…………………………………………………22
The Cross Product of Two Vectors in Space ………………………………..30
Lines and Planes in Space …………………………………………………………..37
Surfaces in Space……………………………………………………………………….50
Cylindrical and Spherical Coordinates …………………………………………57 Review Exercises …………………………………………………………………………………………….68
Problem Solving ……………………………………………………………………………………………..76
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y 3 y y 3 2 C H A P T E R 1 1 Vectors and the Geometry of Space
Section 11.1 Vectors in the Plane 1. (a) v (b) 5 1, 4 2 4, 2 7. u v 6 0, 2 3 9 3, 5 10 6, 5 6, 5 5 4 u v (4, 2) 8. u 11 4 , 4 1 15, 3 2 1 v x v u 25 0, 10 13 v 15, 3 1 2 3 4 5 9. (b) v 5 2, 5 0 3, 5 2. (a) v (b) 3 3, 2 4 y 0, 6 (c) v (a), (d) 3i 5j y -3 -2 -1 -1 1 2 3 x 5 4 (3, 5) (5, 5) -2 -3 v 3 2 v -4 -5 1 (2, 0) x -6 (0, – 6) -1 -1 1 2 3 4 5 3. (a) v 4 2, 3 3 6, 0 10. (b) v 3 4, 6 6 1, 12 (b) 4 (c) v (a), (d) i 12 j y (-1, 12) -8 (-6, 0) -6 -4 v -2 2 x v 8 6 (3, 6) -2 4 2 -4 -8 – 6 – 4 -2 2 6 8 10 x – 4 4. (a) v 1 2, 3 1 3, 2 – 6 (4, -6) (b) 11. (b) v (c) v 6 8, 1 3 2i 4 j 2, 4 (- 3, 2) 2 (a), (d) y 6 v 1 4 2 (8, 3) v -3 -2 -1 x -4 -2 2 4 (6, – 1) 8 x 5. u v u
6. u v u 5 3, 6 2 3 1, 8 4 v
1 4 , 8 0 7 2, 7 1 v 2, 4 2, 4
5, 8 5, 8 (-2, -4) -6 © 2010 Brooks/Cole, Cengage Learning
3 6 y y 3 6 6 2 2 2 2 7 2 v 3 3 5 y 3 2 3 y 2 3 2 3 y Section 11.1 Vectors in the Plane 12. (b) v 5 0, 1 4 5, 3 17. (a) 2 v 2 3, 5 6, 10 (c) v 5i 3j (a) and (d). (- 5, 3) 4 y 10 8 6 y (3, 5) (6, 10) v 2 4 v 2v 2 x -6 -4 (- 5, -1) -2 -2 2 -2 -2 2 4 6 8 10 x (0, – 4) (b) 3v 9, 15 13. (b) v 6 6, 6 2 0, 4 (3, 5) (c) v 4 j 3 v x (a) and (d). (6, 6) -15 -12 -9 -6 -3 -3v -6 -9 (-9, – 15) -12 4 (0, 4) v -15 2 (6, 2) (c) 7 v 21, 35 x y 2 4 6 18 (21, 35 ( 14. (b) v (c) v 3 7, 1 1 10i 10, 0 15 12 9 6 (3, 5) (a) and (d). y 3 v x 3 -3 -3 3 6 9 12 15 18 2 (-10, 0) v 1 x (d) 2 v 2, 10 -8 -6 -4 -2 (-3, -1) -2 -3 2 4 6 8 (7, -1) 4 (3, 5) 3 v (2, 10 ( 15. (b) v 1 2 3 , 3 4 3 1, 5 2 1 2 3 v (c) v i 5 3 j -1 -1 1 2 3 4 5 x (a) and (d) 3 ( 1 , 3( (- 1, 5 ( 2 v
( 3 , 4 ( – 2 -1 1 2 x 16. (b) v (c) v 0.84 0.12, 1.25 0.60 0.72i 0.65 j 0.72, 0.65 (a) and (d). 1.25 1.00 0.75 0.50 0.25 (0.12, 0.60) (0.84, 1.25)
(0.72, 0.65)
v x 0.25 0.50 0.75 1.00 1.25 © 2010 Brooks/Cole, Cengage Learning
4 y u 2 2 2 3 3 2 y 3 3 3 v 1 1 x 2 y 2 y -1 Chapter 11 Vectors and the Geometry of Space 18. (a) 4 v 4 2, 3 8, 12 21. y (- 8, 12) 12 10 4v 8 6 4 u – v – v x (-2, 3) v x -8 – 6 – 4 -2
(b) 1 v 2
1, 3 4 6 22. y u + 2v y (-2, 3) 3 2 2v v x u x -3 -2 -1 – 1 v 3 -2 -3 (1, – 3( 23. (a) 2 u 2 3 4, 9 8 , 6 (b) v u 2, 5 4, 9 2, 14 (c) 0 v 0, 0 (c) 2u 5v 2 4, 9 5 2, 5 18, 7 (-2, 3) 24. (a) 2 u 2 3 3, 8 2, 16 2 (b) v u 8, 25 3, 8 11, 33 -3 -2 -1 0v (c) 2u 5v 2 3, 8 5 8, 25 34, 109 -1 (d) 6u 12, 18 25. v 3 2 2i j 3i 3 j 3, 3 y (- 2, 3) 1 -6 v -2 2 6 10 14 x 2 3 x -6 -10 -14 – 6v -1
-2 3 2 u u -18 (12, -18) -3 19. 26. v 2i j i 2 j 3i j 3, 1 y – u 2 x 1 w v x 1 2 3 u 20. Twice as long as given vector u. y
u 2u
x
© 2010 Brooks/Cole, Cengage Learning
5 y 4 § 3 · § 5 · © 2 ¹ © 2 ¹ v 2 © 2 ¹ © 2 ¹ v 2 2 2 2 Section 11.1 Vectors in the Plane 27. v 2i j 2 i 2 j 4i 3j 4, 3 38. v 5, 15 v 25 225 250 5 10 2w u v v 5, 15 5 10
10 3 10 , 10 10 unit vector 2 u + 2w u 4 6 x 39. v 3 5 , 2 2 -2
28. v 5u 3w 5 2, 1 3 1, 2 7, 11 2 ¨ ¸ ¨ ¸ 34 2 -4 -2 2 y 4 6 8 10 x u v v § 3 · § 5 · ¨ ¸, ¨ ¸ 34 3 34 , 5 34 -3w 5u 2 -6 -8 -10 -12 40. v 6.2, 3.4 3 34 5 34 , 34 34 unit vector 29. u1 4 u2 2 1 3 u1 u2 Q 3 5 3, 5 v
u v v 6.2 3.4 6.2, 3.4 5 2
50 5 2
31 2 17 2 , 50 50 unit vector 30. u1 5 u2 3 4 9 u1 u2 9 6 41. u 1, 1 , v 1, 2 Q 9, 6 Terminal point (a) u 1 1 2 31. v 0 72 7 (b) v 1 4 5 (c) u v 0, 1 32. v 3 2 0 3 u v 0 1 1 33.
34. v
v 42 32
122 5 2 5 13 (d) u u
u u 1 1 2 1, 1 35. v 62 5 2 61 (e) v v 1 5 1, 2 36. v 10 2 32 109 v v 1 37. v
v 3, 12
3 12 153 (f ) u v u v 0, 1 u v v 3, 12 153 3 153 , 12 153 u v u v 1 17 4 17 , 17 17 unit vector © 2010 Brooks/Cole, Cengage Learning
6 1 x 6¨ 6 0, 1 © ¹ Chapter 11 Vectors and the Geometry of Space 42. u 0, 1 , v 3, 3 44. u 2, 4 , v 5, 5 (a)
(b)
(c) u
v u v 0 1 9 9 3, 2 1
3 2 (a)
(b)
(c) u
v u v 4 16 25 25 7, 1 2 5
5 2 u v 9 4 13 u v 49 1 5 2 (d) u u 0, 1 (d) u u 1 2 5 2, 4 u u 1 u u 1 (e) v v 1 3 2 3, 3 (e) v v 1 5 2 5, 5 v v 1 v v 1 (f ) u v u v 1 13 3, 2 (f ) u v u v 1 5 2 7, 1 u v u v 1 u v u v 1 43. u 1, 1 2 , v 2, 3 45. u u 2, 1 5 | 2.236 7 y (a)
(b) u
v 1 4 4 9 5 2 13 v v u v 5, 4 41 | 6.403 7, 5 6 5 4 3 2 1 v u u + v (c) u v
u v 7 3, 2
9 49 4 85 2 u v 74 | 8.602 u v d u v 74 d 5 41 -1 1 2 3 4 5 6 7 (d)
(e) u u
u u
v v
v v 1
1 2 5
1 13 1 1, 2
2, 3 46. u 3, 2 u 13 | 3.606 v 1, 2 v 5 | 2.236 u v 2, 0 u v 2 u+ v d u v u
u+v -3 -2 -1 3 2 1
-1 -2 -3 y 1 v 2 3 x (f ) u v u v u v u v 1 2 85 3, 7 2 47. 2 d 13
u 1 0, 3 0, 1 u 3 § u · ¨ u ¸ 5
0, 6 v 0, 6 © 2010 Brooks/Cole, Cengage Learning
7 4¨ © ¹ ¬ ¼ ¬ ¼ 5¨ © ¹ 1 S , , 2¨ © ¹ 3i 52. v ¬ ¼ ¬ ¼ 5 , j a ¬ ¼ ¬ ¼ u v j ¸i ¨ © ¹ , j u 3 3 v u v Section 11.1 Vectors in the Plane 48. u u § u · ¨ u ¸ 1 1, 1 2
2 2 1, 1 58. u
v u v 5ªcos 0.5 º i 5ªsin 0.5 º j 5 cos 0.5 i 5 sin 0.5 j 5 cos 0.5 i 5 sin 0.5 j 10 cos 0.5 i 10 cos 0.5, 0 v 2 2, 2 2 59. Answers will vary. Sample answer: A scalar is a real 49. u u § u · ¨ u ¸ 1 5 1, 2
5 5
2 5 1 5 2 5
5, 2 5 number such as 2. A vector is represented by a directed line segment. A vector has both magnitude and direction. For example 3, 1 has direction and a magnitude 6 of 2. v
5, 2 5 60. See page 766: (ku1, ku2) 50. u u 1 2 3 3, 3 (u1, u2) (u1 + v1, u2 + v2) u + v ku ku2 § u · ¨ u ¸ 1 3 3, 3 u (v1, v2) v u2
v2 u1 u (u1, u2) u2 v 1, 3 v1 u1 ku1 51. v 3ª cos 0q i sin 0q jº
5ª cos 120q i sin 120q jº 3, 0 61. (a) Vector. The velocity has both magnitude and direction. (b) Scalar. The price is a number. i 2 5 3 2 5 5 3 2 2 62. (a) Scalar. The temperature is a number. (b) Vector. The weight has magnitude and direction. 53. v 2ª cos 150q i sin 150q jº 3i j 3, 1 For Exercises 63–68, au bw a i 2j b i j b i 2a b j. 54. v 4ª cos 3.5q i sin 3.5q jº 63. v 2i j. So, a b 2, 2a b simultaneously, you have a 1, b 1. Solving 1. | 3.9925i 0.2442 j 3.9925, 0.2442 64. v 3j. So, a b 0, 2a b 3. Solving simultaneously, you have a 1, b 1. 55. cos 0q i sin 0q j i 3 cos 45q i 3 sin 45q j 3 2 2 i 3 2 2 65. v 3i. So, a b 3, 2a b simultaneously, you have a 1, b 0. Solving 2. u v § 2 3 2 · ¨ 2 ¸ 3 2 2 2 3 2 3 2 2 2 66. v 3i 3j. So, a b simultaneously, you have a 3, 2a b 2, b 1. 3. Solving 56. 4 cos 0q i 4 sin 0q j 4i 67. v i j. So, a b 1, 2a b 1. Solving 2 cos 30q i 2 sin 30q j i 3j simultaneously, you have a 2 , b 1 . u v 5i 3j 5, 3 68. v i 7 j. So, a b 1, 2a b 7. Solving 57. 2 cos 4 i 2 sin 4 j simultaneously, you have a 2, b 3. cos 2 i sin 2 j u v 2 cos 4 cos 2 i 2 sin 4 sin 2 j 2 cos 4 cos 2, 2 sin 4 sin 2 © 2010 Brooks/Cole, Cengage Learning
8 6 y 8 6 x 6 y 4 3 y 2 1 1 y 4 x y 3 1 2 1 y 2 4 x 1 Chapter 11 Vectors and the Geometry of Space 69. f x x 2 , f c x 2 x, f c 3 10 (a) m 6. Let w 1, 6 , w 37, then r w w r 1 37 1, 6 . (a) 4 (b) (b) m 1 . Let w 6, 1 , w 37, then r w w r 1 37 6, 1 . -2 2 2 (3, 9) 4 6 8 10 70. f x x 2 5, f c x 2 x, f c 1 2 (a) m 2. Let w 1, 2 , w 5, then r w w r 1 5 1, 2 . (a)
(1, 4) (b) (b) m 1 2 . Let w 2, 1 , w 5, then r w w r 1 5 2, 1 . -3 -1 2 1
-1 1 2 3 x 71. f x x3 , f c x 3x 2 3 at x 1. (a) m 3. Let w 1, 3 , w 10, then w w r 1 10 1, 3 . (a) (b) m . Let w 3 3, 1 , w 10, then w w r 1 10 3, 1 . (1, 1) (b) x 1 2 72. f x x3 , f c x 3x 2 12 at x 2. (a) m 12. Let w 1, 12 , w 145, then w w r 1 145 1, 12 . -6 -4 -2 -4 2 (a) (b) m
1 12 . Let w 12, 1 , w 145, then w w r 1 145 12, 1 . -6 (b) -10 73. f x 25 x 2 f c x x 25 x 2 3 4 at x 3. 4 3 (a) (3, 4) (b) (a) m . Let w 4 4, 3 , w 5, then w w r 4, 3 . 5 (b) m 4 3 . Let w 3, 4 , w 5, then w w r 1 5 3, 4 . -1 1 2 3 4 5 x 74. f x tan x f c x sec2 x 2 at x S 4 2.0 1.5 (a) (a) m 2. Let w 1, 2 , w 5, then w w r 1 5 1, 2 . – p – p 1.0 0.5 p 4 (b)
p 2 (b) m . Let w 2 2, 1 , w 5, then w w r 1 5 2, 1 . -1.0 © 2010 Brooks/Cole, Cengage Learning
9 2 j y 1 2 6 x R R 81. 2 2 § 90 · © 430.88 ¹ 2 arctan « » D Section 11.1 Vectors in the Plane 75. u
u v i 2 2 j 2 2 78. (a) v (b) v (c) 9 3, 1 4 6i 5 j 6, 5 v u v u
2 2 i 2 2 j
2 2 , 2 2 6 5 4 (6, 5) 3 v 76. u 2 3i 2 j 2 1 u v v 3i 3 3j u v u 3 2 3 i 3 3 2 j (d) -1
v 3 4 5
62 52 61 3 2 3, 3 3 2
77. (a)–(c) Programs will vary. (d) Magnitude | 63.5 79. F1 F2 F3 2, TF1 3, TF2 2.5, TF3 33q 125q 110q Direction | 8.26q T R F1 F2 F3 | 1.33 TF1 F2 F3 | 132.5q 80. F1 F2 F3 2, TF1 4, TF2 3, TF3 10q 140q 200q F1 F2 F3 | 4.09 T R TF1 F2 F3 | 163.0q F1 F2 500 cos 30qi 500 sin 30q j 200 cos 45q i 200 sin 45q j 250 3 100 2 i 250 100 2 j F1 F2 250 3 100 2 250 100 2 | 584.6 lb tan T 250 100 2 250 3 100 2 ? T | 10.7q 82. (a) 180 cos 30qi sin 30q j 275i | 430.88i 90 j Direction: D | arctan¨ ¸ | 0.206 | 11.8q Magnitude: 430.882 902 | 440.18 newtons (b) M 275 180 cos T 2 180 sin T ª 180 sin T º ¬ 275 180 cos T ¼ (c) T M D 0q 455 0q 30q 440.2 11.8q 60q 396.9 23.1q 90q 328.7 33.2q 120q 241.9 40.1q 150q 149.3 37.1q 180q 95 0 (d) 500 M 50 a 0 0 180 0 0 180 (e) M decreases because the forces change from acting in the same direction to acting in the opposite direction as T increases from 0q to 180q.
© 2010 Brooks/Cole, Cengage Learning
10 125 125 75 3 50 2 i 75 50 2 2 2 2 2 R ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ 2 R 2 arctan¨ T R q. © ¹ S . 3 P 1 P2 Chapter 11
83. F1 F2 F3
T R
84. F1 F2 F3 Vectors and the Geometry of Space
75 cos 30qi 75 sin 30q j 100 cos 45qi 100 sin 45q j 125 cos 120qi 125 sin 120q j 3 j F1 F2 F3 | 228.5 lb TF1 F2 F3 | 71.3q ª400 cos 30q i sin 30q j º ª280 cos 45q i sin 45q j º ª350 cos 135q i sin 135q j º ª200 3 140 2 175 2 ºi ª 200 140 2 175 2 º j 200 3 35 2 200 315 2 | 385.2483 newtons § 200 315 2 · ¨ 200 3 35 2 ¸ | 0.6908 | 39.6q
85. (a) The forces act along the same direction. T (b) The forces cancel out each other. T 180q. (c) No, the magnitude of the resultant can not be greater than the sum. 86. F1
(a) 20, 0 , F2 F1 F2 10 cos T sin T 20 10 cos T , 10 sin T 400 400 cos T 100 cos2 T 100 sin 2 T 500 400 cos T (b) 0 40 2 0 (c) The range is 10 d F1 F2 d 30. The maximum is 30, which occur at T The minimum is 10 at T (d) The minimum of the resultant is 10. 87. 4, 1 , 6, 5 , 10, 3 0 and T 2S . 8 y 8 y 8 y 6 4 (1, 2) (8, 4) 6 4 (1, 2) (6, 5) (8, 4) 6 4 (1, 2) (8, 4) 2
(- 4, -1) 2 4 (3, 1) 6 8 x 2 -4 -2 -2 2 4 (3, 1) 6 8 x 2 -2 -2 (3, 1) 4 6 8 (10, 3) x 10 -4 -4 -4 88. u 1 u 7 1, 5 2 2, 1 6, 3 1, 2 2, 1 3, 3 1, 2 2 2, 1 5, 4
© 2010 Brooks/Cole, Cengage Learning
11 CB CA JJJ JJJ G G y A u v 0 u ¨ © ¹ 100. 0 or § 24 · T2 § 24 · © 10 ¹ u v § 1 · 0. © 2 ¹ 3 and adding to the § 1 · 0 gives © 2 ¹ 5000 0 v v u v § 547.64 · © 692.53 ¹ 2 2 Section 11.1 Vectors in the Plane 89. u v 50° u cos 30qi sin 30q j v cos 130qi sin 130q j
130° 30° B 91. Horizontal component
Vertical component v cos T 1200 cos 6q | 1193.43 ft sec
v sin T 1200 sin 6q | 125.43 ft sec v C u 30° x 92. To lift the weight vertically, the sum of the vertical components of u and v must be 100 and the sum of the horizontal components must be 0. u cos 60qi sin 60q j Vertical components: u sin 30q v sin 130q 3000 v cos 110qi sin 110q j Horizontal components: u cos 30q v cos 130q So, u sin 60q v sin 110q 100, or Solving this system, you obtain u | 1958.1 pounds v | 2638.2 pounds
90. T1 arctan¨ ¸ | 0.8761 or 50.2q © 20 ¹ arctan¨ ¸ S | 1.9656 or 112.6q u cos T1 i sin T1 j v cos T 2 i sin T 2 j Vertical components: u sin T1 v sin T2 § 3 · ¨ 2 ¸ v sin 110q And u cos 60q v cos 110q
u ¨ ¸ v cos 110q
Multiplying the last equation by first equation gives u sin 110q 3 cos 110q 100 ? v | 65.27 lb.
Then, u ¨ ¸ 65.27 cos 110q u | 44.65 lb. Horizontal components: u cos T1 v cos T 2 (a) The tension in each rope: u 44.65 lb, Solving this system, you obtain u | 2169.4 and v | 3611.2. y 65.27 lb (b) Vertical components: u sin 60q | 38.67 lb, v sin 110q | 61.33 lb A v C ?2
?1 u B x 20°
v 30°
u 100 lb 93. u 900 cos 148q i sin 148q j 100 cos 45q i sin 45q j 900 cos 148q 100 cos 45q i 900 sin 148q 100 sin 45q j | 692.53i 547.64 j T | arctan¨ ¸ | 38.34q; 38.34q North of West u v | 692.53 547.64 | 882.9 km h © 2010 Brooks/Cole, Cengage Learning
12 u v v ¸i j ¨ y c u 2 a x u v 1 x 2 ¬ ¼ ¬ ¼ u ¸i sin¨ v «cos¨ u ¸ cos¨ ¸ cos¨ ¸ j 2 u ¬ ¼ ¼ § T © § T © ¸ cos¨ ¸ cos¨ ¸ ¸ tan¨ u © ¸ Chapter 11 Vectors and the Geometry of Space 94. u 400i plane
400 25 2 i 25 2 j | 364.64i 35.36 j 50 cos 135qi sin 135q j 25 2i 25 2 j wind tan T 35.36 364.64 ? T | 5.54q Direction North of East: | N 84.46q E Speed: | 336.35 mi h 95. True
96. True
97. True 102. Let the triangle have vertices at 0, 0 , a, 0 , and b, c . Let u be the vector joining 0, 0 and b, c , as indicated in the figure. Then v, the vector joining the midpoints, is 98. False a
99. False b 0 v § a b a · c © 2 2 ¹ 2 b c i + j 2 2 (b, c) ( a + b , 2 ( ai bj
100. True 2 a 1 2 bi cj 1 2 u. (0, 0) v ( 2 , 0 ( (a, 0) 101. cos2 T sin 2 T sin 2 T cos2 T 1, 103. Let u and v be the vectors that determine the parallelogram, as indicated in the figure. The two diagonals are u v and v u. So, r x u v , s 4 v u . But, r s u x u v y v u y u x y v. So, x y 1 and x y 0. Solving you have 1 . x y s u r v 104. w u v v u u ª v cos T v i v sin T v jº v ª u cos Tu i u sin Tu jº v ª cos Tu cos T v i sin Tu sin T v jº ª § T T v · § Tu T v · § Tu T v · § Tu T v · º ¬ © 2 ¹ © 2 ¹ © 2 ¹ © 2 ¹ » tan T w sin¨ u
cos¨ u T v · § Tu 2 ¹ © T v · § Tu 2 ¹ © T v · 2 ¹ T v · 2 ¹ § T T v · 2 ¹ So, T w Tu T v 2 and w bisects the angle between u and v. 105. The set is a circle of radius 5, centered at the origin. u x, y x 2 y 2 5 ? x 2 y 2 25 © 2010 Brooks/Cole, Cengage Learning
13 gt . y § ¸ g ¨ ¸ t 2 g 2 a 2 gx 2 2v0 x 2v0 2v0 2 g 2 g v0 2 2v0 2 g v0 2 2 gx 2 gx 2 § v 2 · 2 gx 2 2v0 gt 2 v2 § g · v 2 2 g © v0 ¹ 2 g 2 2v0 2v0 2 g v0 2 3 z 4 2 6 z 2 Section 11.2 Space Coordinates and Vectors in Space 106. Let x v0t cos D and y v0t sin D 1 2 2 x v0 cos D ? y v0 sin D ¨ © x · 1 § x · v0 cos D ¹ 2 © v0 cos D ¹ x tan D
x tan D 2v0 2 x sec2 D
1 tan 2 D (x, y) gx 2 gx 2 v 2 2 2 tan 2 D x tan D 0 gx 2 gx 2 ª § v 2 · v 4 º 2 2 «tan 2 D 2 tan D ¨ 0 ¸ 20 2 » 2v0 ¬ © gx ¹ g x ¼ v0 2 g 2 2 ¨ tan D 0 ¸ 2v0 2v0 © gx ¹ 2 If y d v0 2 g 2 , then D can be chosen to hit the point x, y . To hit 0, y : Let D 90q. Then y v0t 1 2 2 v0 2 g 2 0 ¨ t 1¸ , and you need y d 0 . The set H is given by 0 d x, 0 y and y d v0 2 g gx 2 2 Note: The parabola y gx 2 2 is called the “parabola of safety.” Section 11.2 Space Coordinates and Vectors in Space 1. A 2, 3, 4 B 1, 2, 2 2. A 2, 3, 1 B 3, 1, 4 5. (5, – 2, 2) – 3 3 4 x (5, – 2, – 2) 2 1 3 2 1
– 2 – 3 z 1 2 3 y 3. 6 z 6. z 5 4 8 (2, 1, 3) (-1, 2, 1) 6 (4, 0, 5) 2 1 x 4 3 2 2 3 4 y x 6 – 2 – 4 6 y – 6 (0, 4, – 5) 4. 8 7. x 3, y 4, z 5: 3, 4, 5 (3, -2, 5) 6 8. x 7, y 2, z 1: 7, 2, 1 x 6 y ( 3 , 4, -2( 9. y 0, x 12: 12, 0, 0 10. x 0, y 3, z 2: 0, 3, 2 © 2010 Brooks/Cole, Cengage Learning
14 4. 2 2 35. ¨ ¸ , , 2 2 § 3 · © 2 ¹ 2 2 , , © 2 ¹ 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 25 2 2 2 2 2 2 2 2 2 Chapter 11 Vectors and the Geometry of Space 11. The z-coordinate is 0. 30. A 3, 4, 1 , B 0, 6, 2 , C 3, 5, 6 12. The x-coordinate is 0. 13. The point is 6 units above the xy-plane. 14. The point is 2 units in front of the xz-plane. AB AC BC 9 4 1 0 1 25 9 1 16 14 26 26 15. The point is on the plane parallel to the yz -plane that Because AC BC , the triangle is isosceles. passes through x 3. 16. The point is on the plane parallel to the xy-plane that 31. A 1, 0, 2 , B 1, 5, 2 , C 3, 1, 1 passes through z 5 2. AB 0 25 16 41 17. The point is to the left of the xz-plane. 18. The point is in front of the yz-plane. AC
BC 4 1 9 4 36 1 14
41 19. The point is on or between the planes y y 3. 3 and Because AB BC , the triangle is isosceles. 20. The point is in front of the plane x 32. A 4, 1, 1 , B 2, 0, 4 , C 3, 5, 1 21. The point x, y, z is 3 units below the xy-plane, and below either quadrant I or III. 22. The point x, y, z is 4 units above the xy-plane, and above either quadrant II or IV. AB AC BC Neither 4 1 9 1 36 0 1 25 9 14 37 35 23. The point could be above the xy-plane and so above quadrants II or IV, or below the xy-plane, and so below quadrants I or III. 24. The point could be above the xy-plane, and so above quadrants I and III, or below the xy-plane, and so below quadrants II or IV. 33. The z-coordinate is changed by 5 units: 0, 0, 9 , 2, 6, 12 , 6, 4, 3
34. The y-coordinate is changed by 3 units: 3, 7, 1 , 0, 9, 2 , 3, 8, 6 25. d 4 0 2 0 7 0 16 4 49 69 § 5 2 9 3 7 3 · © 2 ¹ ¨ , 3, 5¸ 26. d 2 2 5 3 2 2 § 4 8 0 8 6 20 · 36. ¨ ¸ 6, 4, 7 16 64 16 96 4 6 37. Center: 0, 2, 5 27. d 6 1 2 2 2 4 25 0 36 61 Radius: 2 x 0 y 2 z 5 28. d 4 2 2 5 2 6 3 38. Center: 4, 1, 1 4 49 9 62 29. A 0, 0, 4 , B 2, 6, 7 , C 6, 4, 8 Radius: 5 x 4 y 1 z 1 AB
AC 22 62 32 62 42 12 49 7
196 14 39. Center:
Radius: 2, 0, 0 0, 6, 0 2 10 1, 3, 0 BC BC 4 2 15 245 49 196 2 245 AB AC 7 5 x 1 y 3 z 0 10 Right triangle
© 2010 Brooks/Cole, Cengage Learning
15 2 2 2 2 2 2 2 § 2 81 · 2 2 © 4 ¹ § 9 · 2 © 2 ¹ 2 § 9 · © 2 ¹ 2 2 2 3 3 9 2 2 2 3 0 3 2 2 2 40. Center: 3, 2, 4 Section 11.2 Space Coordinates and Vectors in Space r 3 tangent to yz-plane x 3 y 2 z 4 2 9 41. x x 2 y 2 z 2 2 x 6 y 8 z 1 2 x 1 y 6 y 9 z 8 z 16 0 1 1 9 16 x 1 y 3 z 4 2 25 Center: 1, 3, 4 Radius: 5 42. x 2 y 2 z 2 9 x 2 y 10 z 19 0 ¨ x 9 x ¸ y 2 y 1 z 10 z 25 2 ¨ x ¸ y 1 z 5 19
109 4 81 4 1 25 Center: ¨ , 1, 5¸ Radius: 109 2 43. 9 x 2 9 y 2 9 z 2 6 x 18 y 1 0 x y z 2 x 2 y 1 9 0 x 2
2 x
1 9 y 2 2 y 1 z 2 1 1 9 1 x 1 y 1 z 0 1 Center: 1 , 1, 0 Radius: 1
44. 4 x 2 4 y 2 4 z 2 24 x 4 y 8 z 23 x 2 6 x 9 y y 1 4 z 2 2 z 1 23 4 9 1 4 1 x 3 y 1 2
2 z 1 2 16 Center: 3, 1 , 1 Radius: 4
45. x 2 y 2 z 2 d 36 Solid sphere of radius 6 centered at origin.
46. x 2 y 2 z 2 ! 4 Set of all points in space outside the ball of radius 2 centered at the origin.
© 2010 Brooks/Cole, Cengage Learning
16 47. 2 2 2 x 48. 2 2 2 x , , 5 4 3 2 1 z , , z 8 6 4 2 , 3 2 v Chapter 11 Vectors and the Geometry of Space
x 2 y 2 z 2 4 x 6 y 8 z 13 x2 4 x 4 y 2 6 y 9 z 2 8 z 16 4 9 16 13 2 y 3 z 4 16 Interior of sphere of radius 4 centered at 2, 3, 4 .
x 2 y 2 z 2 ! 4 x 6 y 8 z 13 x2 4 x 4 y 2 6 y 9 z 2 8 z 16 ! 13 4 9 16 2 y 3 z 4 ! 16 Set of all points in space outside the ball of radius 4 centered at 2, 3, 4 . 49. (a) v 2 4, 4 2, 3 1 2, 2, 2 53. 4 3, 1 2, 6 0 1, 1, 6 (b) v 2i 2 j 2k 1, 1, 6 1 1 36 38 (c) -2 – 3 < – 2, 2, 2> 54. 1, 1, 6 Unit vector: 38
1 4, 7 5 , 3 2 1 38 1 6 38 38
5, 12, 5 3 2 1 1 2 3 4 y 5, 12, 5 25 144 25 194 x
50. (a) v 4 0, 0 5, 3 1 4, 5, 2 Unit vector: 5, 12, 5 194 5 194 12 194 5 194 (b) v (c) 4i 5 j 2h 55. 5 4 , 3 3, 0 1 1, 0, 1 1 1 1, 0, 1 2 < 4, – 5, 2 > Unit vector: 1, 0, 1 2 1 2 , 0, 1 2 x 6 4 2 2 4 6 y 56. 2 1, 4 2 , 2 4 1, 6, 6 51. (a) v (b) v (c) 0 3, 3 3, 3 0 3i 3k z < -3, 0, 3> 5 4 -3 3, 0, 3 1, 6, 6 1 36 36 1, 6, 6 1 Unit vector: , 73 73
57. (b) v 3 1 , 3 2, 4 3 4i j k (c) v 73 6 6 73 73
4, 1, 1 -2 2 3 1 1 1 2 3 4 y (a), (d) 5 z (3, 3, 4) x 4 3 (0, 0, 0) 2 (-1, 2, 3) 52. (a) v 2 2, 3 3, 4 0 0, 0, 4 – 2 (b) v 4k (4, 1, 1) 2 4 2 4 y (c) 4 3 2 1 z < 0, 0, 4 > x x 3 2 1 1 2 3 y © 2010 Brooks/Cole, Cengage Learning
17 z z 4 3 z 8 6 4 2 Q 2 x 6 y 3 3 2 2 2 2 5 8 6 4 2 – 2 1 2 3 y x 2 2 3 2 2 2 z 2 2 Section 11.2 Space Coordinates and Vectors in Space 58. (b) v 4 2, 3 1 , 7 2 6, 4, 9 62. (a) v 2, 2, 1 (c) v (a), (d) 6i 4 j 9k
(- 4, 3, 7) 12 9 6 (- 6, 4, 9) 2 1 3 x 3 3 >- 2, 2, -1> y x 9 9 y (b) 2 v 4, 4, 2 (2, -1, – 2) 59. q1 , q2 , q3 0, 6, 2 3, 5, 6 Q 3, 1, 8 >4, – 4, 2> 60. q1 , q2 , q3 0, 2, 1, 4 , 3 5 2 1, 2 , 1 (c) 1 v 6
1, 1, 1 2 z 61. (a) 2 v 2, 4, 4 z 5 4 >1, -1, 1 > 1 3 2 < 2, 4, 4> x y 4 -2 2 3 1 1 2 y (d) 5 v 5, 5, 5 2 x z (b) v 1, 2, 2 z
3 < 5, -5, 2 < 2 – 2 – 3 x 6 y – 3 < – 1, -2, -2> 2 3 63. z u v 1, 2, 3 2, 2, 1 1, 0, 4 – 2 – 3 64. z u v 2w 1, 2, 3 2, 2, 1 8, 0, 8 7, 0, 4 (c) 3 v 3 , 3, 3
z 65. z 2u 4v w 2, 4, 6 8, 8, 4 4, 0, 4 6, 12, 6 – 3 -2 -2 – 3 < 3 , 3, 3> 66. z 5u 3v 1 w x 3 2 – 2 – 3 1 y 5, 10, 15 6, 6, 3 2, 0, 2 3, 4, 20 (d) 0v 0, 0, 0 67. 2z 3u 2 z1 3 2 z1 , z2 , z3 3 1, 2, 3 4 ? z1 7 2 4, 0, 4 -3
x -2 2 3 1 3 2 1
-1 -2 -3 – 3 – 2 < 0, 0, 0> 1 2 3 y 2 z2 6 0 ? z2 2 z3 9 4 ? z3 z 7 , 3, 5 3 5 2 © 2010 Brooks/Cole, Cengage Learning
18 0 2 i 3 k 2 1 i 3 k and 3 2 2 3 4 j 9 k 3 k . 4 8 3 i 3 1 i 2 j 3 3 K K K K K KK 2 K K v 2 K K K K K Chapter 11 Vectors and the Geometry of Space 68. 2u v w 3z 2 1, 2, 3 2, 2, 1 4, 0, 4 3 z1 , z2 , z3 0, 0, 0 0, 6, 9 3z1 , 3z2 , 3z3 0, 0, 0 0 3z1 0 ? z1 6 3z2 9 3z3 0 ? z2 0 ? z3 2 3 z 0, 2, 3 69. (a) and (b) are parallel because 6, 4, 10 2 3, 2, 5 and 2, 4 , 10 3 3, 2, 5 .
70. (b) and (d) are parallel because 4 j 2 j
2 2 3 4 71. z 3i 4 j 2k (a) is parallel because 6i 8j 4k 2z. 72. z 7, 8, 3 (b) is parallel because z z 14, 16, 6 . 73. P 0, 2, 5 , Q 3, 4, 4 , R 2, 2, 1 JJJK PQ 3, 6, 9 JJJ PR 2, 4, 6 77. A 2, 9, 1 , B 3, 11, 4 , C 0, 10, 2 , D 1, 12, 5 JJJ AB 1, 2, 3 JJJK CD 1, 2, 3 JJJK AC 2, 1, 1 JJJK BD 2, 1, 1 JJJ JJJK JJJK JJJK Because AB CD and AC BD, the given points form the vertices of a parallelogram. 78. A 1, 1, 3 B 9, 1, 2 , C 11, 2, 9 , D 3, 4, 4 JJJ AB 8, 2, 5 JJJK DC 8, 2, 5 JJJK AD 2, 3, 7 JJJ BC 2, 3, 7 JJJ JJJK JJJK JJJ Because AB DC and AD BC , the given points form the vertices of a parallelogram. 3, 6, 9 3 2, 4, 6 JJJK JJJ So, PQ and PR are parallel, the points are collinear.
74. P 4, 2, 7 , Q 2, 0, 3 , R 7, 3, 9 JJJK PQ 6, 2, 4 JJJ PR 3, 1, 2 79. v v
80. v v
81. v 0, 0, 0 0
1, 0, 3 1 0 9 10
3j 5k 0, 3, 5 3, 1, 2 1 6, 2, 4 JJJK JJJ So, PQ and PR are parallel. The points are collinear.
75. P 1, 2, 4 , Q 2, 5, 0 , R 0, 1, 5 JJJK PQ 1, 3, 4 JJJ PR 1, 1, 1 JJJK JJJ Because PQ and PR are not parallel, the points are not collinear. 76. P 0, 0, 0 , Q 1, 3, 2 , R 2, 6, 4 JJJK PQ 1, 3, 2 JJJ PR 2, 6, 4 v
82. v v
83. v v
84. v 0 9 25
2i 5 j k 4 25 1
i 2 j 3k 1 4 9
4i 3j 7k 16 9 49 34
2, 5, 1 30
1, 2, 3 14
4, 3, 7
74 JJJK JJJ Because PQ and PR are not parallel, the points are not collinear.
© 2010 Brooks/Cole, Cengage Learning
19 7 3 2 7 , , , , , ¬ ¼ 1 z y v z 8 6 3 Section 11.2 Space Coordinates and Vectors in Space 85. v 2, 1, 2 91. cv c 2i 2 j k 4c 2 4c 2 c 2 v 4 1 4 3 9c 2 7 (a) v v 1 3 2, 1, 2 9c 2
c 49 r 7 (b) v v
1 3 2, 1, 2 92. cu 14c c i 2 j 3k 4 c 2 4c 2 9c 2 4 86. v 6, 0, 8 14c 2 16 v 36 0 64 10 c r 8 (a) v v 1 10 6, 0, 8 93. v 10 u u 10 0, 3, 3 3 2 (b) v v
1 10 6, 0, 8 10 0, 1 2 1 2 0, 10 10 , 2 2 87. v v 3, 2, 5 9 4 25 38 94. v 3 u u 3 1, 1, 1 3 (a) v v 1 38 3, 2, 5 3 1 3 , 1 3 , 1 3 3 3 , 3 3 , 3 3 (b) v v
1 38 3, 2, 5 95. v 3 u 2 u 3 2, 2, 1 2 3 3 2 2 1 2 3 3 3 1, 1, 1 2 88. v v 8, 0, 0 8 96. v 7 u u 7 4, 6, 2 2 14 14 21 7 14 14 14 (a) v v 1 8 1, 0, 0 97. v 2ªcos r30q j sin r30q k º 3j r k 0, 3, r1 (b) v v 1, 0, 0 8 2 -2 89. (a)–(d) Programs will vary.
(e) u v 4, 7.5, 2 -2 2 -1 1 1
-1 < 0, 3, 1> u v | 8.732 x -2 < 0, 3, – 1> u | 5.099 v | 9.019 98. v 5 cos 45qi sin 45qk 5 2 2 i k or 90. The terminal points of the vectors tu, u tv and 5 cos 135qi sin 135qk 5 2 2 i k su tv are collinear. su + tv 5 2 2 (i + k) 5 2 2 (- i + k) 4 2 su u + tv 6 x 6 y u v tv 99. v 3, 6, 3 2 v 2, 4, 2 4, 3, 0 2, 4, 2 2, 1, 2 © 2010 Brooks/Cole, Cengage Learning
20 2 2 3 3
3 2 2 2 4 3 z 1 v 0 8 h 8 8 L2 182 8L T T 2 2 2 2 2 2 2 2 0 K K K K K K Chapter 11 Vectors and the Geometry of Space 100. v 5, 6, 3 108. r r0 x 1 y 1 z 1 2 2 1, 2, 5 10 , 2 v 4, 2 10 , 4, 2 13 , 6, 3 x 1 y 1 z 1 This is a sphere of radius 2 and center 1, 1, 1 . 101. (a) 109. (a) The height of the right triangle is h JJJK The vector PQ is given by JJJK PQ 0, 18, h . L2 182 . x 1 u 1 y The tension vector T in each wire is (b) w au bv ai a b j bk T c 0, 18, h where ch 24 3 8. a 0, a b 0, b 0 So, a and b are both zero. So, T 0, 18, h and (c) ai a b j bk a 1, a b 2, b w u v (d) ai a b j bk a 1, a b 2, b Not possible i 2 j k 1
i 2 j 3k 3 182 h 2 h 182 L2 182
, L ! 18. L2 182 Q (0, 0, h) 102. A sphere of radius 4 centered at x1 , y1 , z1 . v x x2 y y1 , z z1 L x x1 y y1 z z1 2 4 (0, 18, 0) x x1 y y1 z z1 2 16 (0, 0, 0) 18 P 103. x0 is directed distance to yz-plane. y0 is directed distance to xz-plane. (b) L T 20 18.4 25 11.5 30 10 35 9.3 40 9.0 45 8.7 50 8.6 z0 is directed distance to xy-plane. 104. d x2 x1 y2 y1 z2 z1 2 (c) 30 L = 18 105. x x0 y y0 z z0 2 r 2 T =8 0 100 106. Two nonzero vectors u and v are parallel if u some scalar c. cv for 0 x 18 is a vertical asymptote and y horizontal asymptote. 8 is a 107. B (d) lim L o18 8L L2 182 f C lim L of 8L L2 182 lim L of 8 1 18 L 2 8 A JJJ JJJ JJJK AB BC AC JJJ JJJ JJJ So, AB BC CA JJJK JJJ AC CA (e) From the table, T
110. As in Exercise 109(c), x asymptote. So, lim T r0 o a 10 implies L 30 inches.
a will be a vertical f. © 2010 Brooks/Cole, Cengage Learning
21 K z 69 23 69 x C AB AC K K K F 2 2 2 2 0 § 2 8 © 3 16 · § 2 1 · 2 2 9 ¹ © 9 ¹ 1 9 § 4 · § 1 · 2 © 3 ¹ © 3 ¹ § 4 1 · © 3 3 ¹ Section 11.2 Space Coordinates and Vectors in Space 111. Let D be the angle between v and the coordinate axes. v cos D i cos D j cos D k v 3 cos D 1 JJJ 113. AB JJJK AC JJJK AD 0, 70, 115 , F1 60, 0, 115 , F2 45, 65, 115 , F3 C1 0, 70, 115 C2 60, 0, 115 C3 45, 65, 115 cos D
v 1 3 3 3 3 3 i j k 3 3 1, 1, 1 F So: F1 F2 F3 0, 0, 500 60C2 45C3 0 70C1 65C3 0 115 C1 C2 C3 500 0.6 Solving this system yields C1 104 , C 2 28 , and 0.4 0.2 ( 3 3 , 3 3 , 3 3 ( C3 112 . So: 0.6 0.4 0.2 0.4 y F1 | 202.919 N F2 | 157.909 N F3 | 226.521N 112. 550 302,500 c 2 c 75i 50 j 100k 18,125c 2 16.689655 c | 4.085 F | 4.085 75i 50 j 100k | 306i 204 j 409k 114. Let A lie on the y-axis and the wall on the x-axis. Then A JJJ JJJK AB 8, 10, 6 , AC 10, 10, 6 .
AB 10 2, AC 2 59 JJJ JJJK 420 JJJJK , F2 650 JJJJJ Thus, F1 AB AC 0, 10, 0 , B 8, 0, 6 , 10, 0, 6 and F1 F2 | 237.6, 297.0, 178.2 423.1, 423.1, 253.9 | 185.5, 720.1, 432.1 F | 860.0 lb 115. d AP 2d BP x 2 y 1 z 1 x 1 2 y 2 z 2 x 2 y 2 z 2 2 y 2 z 2 4 x 2 y 2 z 2 2 x 4 y 5 3x 2 3 y 2 3z 2 8 x 18 y 2 z 18 6 16 9 9 ¨ x x ¸ y 6 y 9 ¨ z 3 z ¸ 44 9
Sphere; center: ¨ , 3, ¸, radius: 2 ¨ x ¸ y 3 ¨ z ¸
2 11 3 2 © 2010 Brooks/Cole, Cengage Learning
22 i 2 2 2 2 1 6 2 2 2 2 2 2 2 2 2 Chapter 11 Vectors and the Geometry of Space Section 11.3 The Dot Product of Two Vectors 1. u 3, 4 , v 1, 5 6. u i, v 2
(d) u ? v v 17 1, 5 17, 85 (e) u ? 2v 2 u ? v 2 17 34
(d) u ? v v 22 2, 3 44, 66 (e) u ? 2v 2 u ? v 2 22 44 (a) u ? v 3 1 4 5 17 (b) u ? u 3 3 4 4 25 (c) u 32 42 25
2. u 4, 10 , v 2, 3 (a) u ? v 4 2 10 3 22 (b) u ? u 4 4 10 10 116 (c) u 42 102 116 (a) u ? v 1 (b) u ? u 1 (c) u 1 (d) u ? v v i (e) u ? 2v 2 u ? v 2
7. u 2i j k , v i k (a) u ? v 2 1 1 0 1 1 (b) u ? u 2 2 1 1 1 1 (c) u 22 1 12 6 (d) u ? v v v i k (e) u ? 2v 2 u ? v 2 3. u = 6, 4 , v 3, 2 (a) u ? v 6 3 4 2 26 (b) u ? u 6 6 4 4 52 (c) u 62 4 52 (d) u ? v v 26 3, 2 78, 52 (e) u ? 2v 2 u ? v 2 26 52 8. u 2i j 2k , v i 3j 2k (a) u ? v 2 1 1 3 2 2 (b) u ? u 2 2 1 1 2 2 (c) u 22 12 2 9 (d) u ? v v 5 i 3j 2k (e) u ? 2v 2 u ? v 2 5 5 9
5i 15 j 10k 10 4. u 4, 8 , v 7, 5 9. u ? v u v cos T (a) u ? v (b) u ? u 4 7 8 5 4 4 8 8 12 80 u ? v 8 5 cos S 3 20 (c) u (d) u ? v v (e) u ? 2v 4 2 82 12 7, 5 2 u ? v 80
84, 60 2 12 24 10. u ? v u v
u ? v cos T
40 25 cos 5S 6 500 3 5. u 2, 3, 4 , v 0, 6, 5 11. u 1, 1 , v 2, 2 (a) u ? v (b) u ? u
(c) u 2 0 3 6 4 5 2 2 3 3 4 4 2 3 4 29 2
29 cos T
T u ? v u v S 2 0 2 8 0 (d) u ? v v 2 0, 6, 5 0, 12, 10 12. u 3, 1 , v 2, 1 (e) u ? 2v 2 u ? v 2 2 4 cos T u ? v u v 5 10 5 1 2 T S 4 © 2010 Brooks/Cole, Cengage Learning
23 3 2 , 6 § 1 · arccos¨ ¸ | 98.1q 1 , 2 3 2 T § S · § S · © 6 ¹ © 6 ¹ § 3S · § 3S · © 4 ¹ © 4 ¹ 1 2 u v j 3 6 ¨ 2 © ¹ © ¹ 3
T cos T S 0 0 arccos¨ © ¹ T cos T T arccos¨ © ¹ 5 3 5 AC CA BC CB 3 5 3 3 21 14 2 2 K K K 2 K 2 2 2 2 K K K 2 K K Section 11.3 The Dot Product of Two Vectors 13. u 3i j, v 2i 4 j 20. u 2, 18 , v 1 cos T u ? v u v 2 10 20 1 5 2 u z cv ? not parallel u ? v 0 ? orthogonal © 5 2 ¹ 21. u 4, 3 , v u z cv ? not parallel 14. cos¨ ¸i sin¨ ¸ j
cos¨ ¸i sin¨ ¸ j 3 2
i j 2
i 2 2 2 u ? v
22. u u 0 ? orthogonal 1 i 2 j , v 2i 4 j 1 v ? parallel cos T u ? v u v 3 § 2 · 1 § 2 · ¨ 2 ¸ 2 ¨ 2 ¸ 2 4 1 23. u j 6k , v i 2 j k u z cv ? not parallel u ? v 8 z 0 ? not orthogonal Neither ª 2 arccos « ¬ 4 1 º 3 » ¼ 105q 24. u 2i 3j k , v u z cv ? not parallel 2i j k 15. u 1, 1, 1 , v 2, 1, 1 u ? v 0 ? orthogonal cos T u ? v u v 2 3 6 2 3 25. u 2, 3, 1 , v 1, 1, 1 u z cv ? not parallel u ? v 0 ? orthogonal T arccos 2 3 | 61.9q 26. u cos T , sin T , 1 , 16. u 3i 2 j k , v u ? v u v
T = 2 2i 3j 3 2 2 3 0 u v v sin T , cos T , 0 u z cv ? not parallel u ? v 0 ? orthogonal 27. The vector 1, 2, 0 joining 1, 2, 0 and 0, 0, 0 is perpendicular to the vector 2, 1, 0 joining 17. u 3i 4 j, v = 2 j 3k 2, 1, 0 and 0, 0, 0 : 1, 2, 0 ? 2, 1, 0 cos T u ? v u v 8 5 13 8 13 65 The triangle has a right angle, so it is a right triangle. 28. Consider the vector 3, 0, 0 joining 0, 0, 0 and § 8 13 · ¨ 65 ¸ | 116.3q
18. u 2i 3j k , v i 2 j k u ? v 9 9 u v 14 6 2 21 § 3 21 · ¨ 14 ¸ | 10.9q
19. u 4, 0 , v 1, 1 u z cv ? not parallel u ? v 4 z 0 ? not orthogonal Neither 3, 0, 0 , and the vector 1, 2, 3 joining 0, 0, 0 and 1, 2, 3 : 3, 0, 0 ? 1, 2, 3 3 0 The triangle has an obtuse angle, so it is an obtuse triangle. 29. A 2, 0, 1 , B 0, 1, 2 , C 1 , 3 , 0 JJJ
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