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Vectors and the geometry of space

Enviado por Efrain Olivo



Partes: 1, 2



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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 © 2010 Brooks/Cole, Cengage Learning

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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