Section 9.7 AN71 9.5 Assess Your Understanding (page 657) 2. dot product 3. orthogonal 4. parallel 5. T 6. F 7. d 8. b 9. (a) 0 (b) 90° (c) orthogonal 11. (a) 0 (b) 90° (c) orthogonal 13. (a) 3 1 − (b) 75° (c) neither 15. (a) 50 − (b) 180° (c) parallel 17. (a) 0 (b) 90° (c) orthogonal 19. 2 3 21. v i j v i j 5 2 5 2 , 1 2 1 2 1 2 = − = − − 23. v i j v i j 1 5 2 5 , 6 5 3 5 1 2 = − − = − 25. v i j v i j 14 5 7 5 , 1 5 2 5 1 2 = + = − 27. − − + i j i j 12 9 or 12 9 29. 9 ft-lb 31. (a) ≈ I 0.022; the intensity of the sun’s rays is approximately 0.022 W/cm .2 = A 500; the area of the solar panel is 500 cm .2 (b) W 10; = ten watts of energy is collected. (c) Vectors I and A should be parallel with the solar panels facing the sun. 33. Force required to keep the Explorer from rolling down the hill: 737.6 lb; force perpendicular to the hill: 5248.4 lb 35. Timmy must exert 85.5 lb. 37. 60° 39. Let a b v i j. = + Then a b 0 v 0 0 0. ⋅ = + = 41. v i j w i j cos sin , 0 ; cos sin , 0 . α α α π β β β π = + ≤ ≤ = + ≤ ≤ If θ is the angle between v and w, then v w cos ,θ ⋅ = since = = v w 1 and 1. Now or . θ α β θ β α = − = − Since the cosine function is even, v w cos . α β ( ) ⋅ = − Also, α β α β ⋅ = + v w cos cos sin sin . So cos cos cos sin sin. α β α β α β ( ) − = + 43. ( ) ( ) + ⋅ − = ⋅ − ⋅ + ⋅ − ⋅ = ⋅ − ⋅ = − = wv vw wv vw wvv wvvw vwwv vww wvv vww w v v w 0 2 2 2 2 2 2 2 2 45. 40 4 507 37 1.353 − + ≈ 47. 2 3, 2 3 − 49. (a) If a b u i j 1 1 = + and a b v i j, 2 2 = + then, since = + = = = + a b a b u v u v , , 1 2 1 2 2 2 2 2 2 2 a a a a b b b b a b a b u v u v 0. 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 2 ( ) ( ) ( )( ) ( )( ) ( ) ( ) + ⋅ −=+ −++ −=+−+= (b) The legs of the angle can be made to correspond to vectors u v + and u v. − 52. 12 53. 9 2 54. 1 sin 1 tan cos sec cos 1 cos 1 2 2 2 2 2 2 θ θ θ θ θ θ ( )( ) ( )( ) − + = = ⋅ = 55. V x x x x V x x x x 19 2 13 2 ,or 4 64 247 3 2 ( ) ( )( ) ( ) = − − = − + 56. { } + + − ln3 ln16 ln7 ln7 ln2 57. f x x 4 9 3 ( ) = + + 58. Vertical asymptotes: x x 3, 5; = − = Horizontal asymptote: y 2 = 59. 3 2 − 60. Vertex: 9, 44; ( ) − concave up 61. [ ] ( ) ( ) [ ] ( ) ( )( ) ( ) = + = + = + = = f g x x x x x x 1 3 tan 9 1 9 tan 9 1 9 tan 1 1 9 sec 1 27 sec 2 3 2 2 3 2 2 3 2 2 3 2 3 9.6 Assess Your Understanding (page 667) 2. components 3. 1 4. F 5. T 6. a 7. All points of the form x z , 0, ( ) 9. All points of the form x y , , 2 ( ) 11. All points of the form y z 4, , ( ) − 13. All points of the form z 1, 2, ( ) 15. 21 17. 33 19. 26 21. 2, 0, 0 ; 2, 1, 0 ; 0, 1, 0 ; 2, 0, 3 ; 0, 1, 3 ; 0, 0, 3 ( ) ( ) ( ) ( ) ( ) ( ) 23. 1, 4, 3 ; 3, 2, 3 ; 3, 4, 3 ; 3, 2, 5 ; 1, 4, 5 ; 1, 2, 5 ( ) ( ) ( ) ( ) ( ) ( ) 25. 1, 2, 2 ; 4, 0, 2 ; 4, 2, 2 ; 1, 2, 5 ; 4, 0, 5 ; 1, 0, 5 ( ) ( ) ( ) ( ) ( ) ( ) − − − 27. v i j k 3 4 = + − 29. v i j k 2 4 = + + 31. v i j 8 = − 33. 7 35. 3 37. 22 39. j k2 − − 41. 105 43. 38 17 − 45. i 47. i j k 3 7 6 7 2 7 − − 49. i j k 3 3 3 3 3 3 + + 51. v w 0; 90 θ ⋅ = = ° 53. v w 2, 100.3 θ ⋅ = − ≈ ° 55. v w 0; 90 θ ⋅ = = ° 57. v w 52; 0 θ ⋅ = = ° 59. α β γ ( ) ≈ ° ≈ ° ≈ ° = ° + ° + ° v i j k 64.6 ; 149.0 ; 106.6 ; 7 cos 64.6 cos 149.0 cos 106.6 61. α β γ ( ) = = ≈ ° = ° + ° + ° v i j k 54.7 ; 3 cos 54.7 cos 54.7 cos 54.7 63. α β γ ( ) = = ° = ° = ° + ° + ° v i j k 45 ; 90 ; 2 cos 45 cos 45 cos 90 65. α β γ ( ) ≈ ° ≈ ° ≈ ° = ° + ° + ° v i j k 60.9 ; 144.2 ; 71.1 ; 38 cos 60.9 cos 144.2 cos 71.1 67. (a) = + + = < > d a b c 7, 1, 5 (b) 8.66 ft 69. x y z 3 1 1 1 2 2 2 ( ) ( ) ( ) − + − + − = 71. Radius 2, center 1, 1, 0 ( ) = − 73. Radius 3, center 2, 2, 1 ( ) = − − 75. Radius 3 2 2 , center 2, 0, 1 ( ) = − 77. 2 newton-meters 2 = joules 79. 9 newton-meters 9 = joules 80. < ≤ ⎧ ⎨ ⎪ ⎩⎪⎪ ⎫ ⎬ ⎪ ⎭⎪⎪ x x 2 13 5 or 2, 13 5 ( ⎤ ⎦ ⎥ 81. x x 2 2 5 2 + − 82. 1 2 83. c A B 3 5 6.71; 26.6 ; 63.4 = ≈ ≈ ° ≈ ° 84. 5 5 85. P x x x x x 2 11 2 10 4 3 2 ( ) = − + − + 86. f x x x 8 5 1 ( ) = + − 87. 24 π 88. 9 18 π − square units 9.7 Assess Your Understanding (page 673) 1. T 2. T 3. T 4. F 5. F 6. T 7. 2 9. 4 11. A B C 11 2 5 − + + 13. A B C 6 23 15 − + − 15. (a) i j k 5 5 5 + + (b) i j k 5 5 5 − − − (c) 0 (d) 0 17. (a) i j k − − (b) i j k − + + (c) 0 (d) 0 19. (a) i j k 2 2 − + + (b) i j k 2 2 − − (c) 0 (d) 0 21. (a) i j k 3 4 − + (b) i j k 3 4 − + − (c) 0 (d) 0 23. i j k 9 7 3 − − − 25. i j k 9 7 3 + + 27. 0 29. i j k 27 21 9 − − − 31. i j k 18 14 6 − − − 33. 0 35. 25 − 37. 25 39. 0 41. Any vector of the form c i j k 9 7 3 , ( ) − − − where c is a nonzero scalar 43. Any vector of the form c i j k5 , ( ) − + + where c is a nonzero scalar 45. 166 47. 555 49. 34 51. 998 53. i j k 11 19 57 19 57 7 19 57 + + or i j k 11 19 57 19 57 7 19 57 − − − 55. 98 cubic units 57. ( ) ( ) ( ) × = = − − − + − a b c a b c b c b c a c a c a b a b u v i j k i j k 1 1 1 2 2 2 1 2 2 1 1 2 2 1 1 2 2 1 ( ) ( ) ( ) × = − + − + − = − + + − + + − + b c b c a c a c a b a b b c b b c c b c a c aacc ac ab aabb ab u v ) 2 2 2 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 a b c a b c u v , 2 1 2 1 2 1 2 2 2 2 2 2 2 2 = + + = + + (continued)
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