648 CHAPTER 9 Polar Coordinates; Vectors In Problems 27–34, the vector v has initial point P and terminal point Q. Find its position vector. That is, express v in the form + ai j. 27. ( ) ( ) = = P Q 0, 0 ; 3, 4 28. ( ) ( ) = = − − P Q 0, 0 ; 3, 5 29. ( ) ( ) = = P Q 3, 2 ; 5, 6 30. ( ) ( ) = − = P Q 3, 2 ; 6, 5 31. ( ) ( ) − − = − P Q 2, 1; 6, 2 32. ( ) ( ) = − = P Q 1, 4 ; 6, 2 33. ( ) ( ) = = P Q 1, 0 ; 0, 1 34. ( ) ( ) = = P Q 1, 1 ; 2, 2 In Problems 35–42, find v . 35. = − v i j 3 4 36. = − + v i j 5 12 37. = − v i j 38. = − − v i j 39. = − + v i j 2 3 40. = + v i j 6 2 41. θ θ = + v i j cos sin 42. θ = + v i j cot In Problems 43–48, find each quantity if = − v i j 3 5 and = − + w i j 2 3 . 43. +v w 2 3 44. −v w 3 2 45. −v w 46. + v w 47. − v w 48. + v w In Problems 49–54, find the unit vector in the same direction as v. 49. = v i5 50. = − v j3 51. = − v i j 3 4 52. = − + v i j 5 12 53. = − v i j 54. = − v i j 2 56. If = v 2, what is the magnitude of − v 3 4 ? 55. If = v 4, what is the magnitude of +v v 1 2 3 ? 57. Find a vector v whose magnitude is 4 and whose component in the i direction is twice the component in the j direction. 58. Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction. 59. If = − v i j 2 and = + x w i j3 , find all numbers x for which + = v w 5. 60. If ( ) = − P 3, 1 and ( ) = Q x, 4 , find all numbers x so that the vector represented by PQ has length 5. In Problems 61–66, write the vector v in the form + a b i j, given its magnitude v and the angle α it makes with the positive x-axis. 61. v 5, 60 α = = ° 62. α = = ° v 8, 45 63. α = = ° v 14, 120 64. α = = ° v 3, 240 65. α = = ° v 25, 330 66. α = = ° v 15, 315 In Problems 67–74, find the direction angle of v. 67. = + v i j 3 3 68. = + j v i 3 69. = − +i v j 3 3 3 70. = − − v i j 5 5 71. = − v i j 4 2 72. = − v i j 6 4 73. = − − v j i 5 74. = − + v j i 3 78. Resultant Force Two forces of magnitude 30 newtons (N) and 70 N act on an object at angles of ° 45 and ° 120 with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find + F F . 1 2 y 1208 458 x iF1i 5 30 N iF2i 5 70 N 79. Finding the Actual Speed and Direction of an Aircraft A Boeing 787 Dreamliner maintains a constant airspeed of 550 miles per hour (mph) headed due north. The jet stream is 100 mph in the northeasterly direction. (a) Express the velocity va of the 787 relative to the air and the velocity vw of the jet stream in terms of i and j. (b) Find the velocity of the 787 relative to the ground. (c) Find the actual speed and direction of the 787 relative to the ground. 75. Force Vectors A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of ° 30 with the ground. Express the force vector F in terms of i and j. 76. Force Vectors A man pushes a wheelbarrow up an incline of ° 20 with a force of 100 pounds. Express the force vector F in terms of i and j. 77. Resultant Force Two forces of magnitude 40 newtons (N) and 60 N act on an object at angles of ° 30 and − ° 45 with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find + F F . 1 2 y 2458 308 x iF1i 5 40 N iF2i 5 60 N Applications and Extensions
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