668 CHAPTER 9 Polar Coordinates; Vectors In Problems 59–66, find the direction angles of each vector. Write each vector in the form of equation (7). 59. v i j k 3 6 2 = − − 60. v i j k 6 12 4 = − + + 61. v i j k = + + 62. v i j k = − − 63. v i j = + 64. v j k = + 65. v i j k 3 5 2 = − + 66. v i j k 2 3 4 = + − Applications and Extensions 67. Robotic Arm Consider the double- jointed robotic arm shown in the figure. Let the lower arm be modeled by a 2, 3, 4 , = the middle arm be modeled by b 1, 1, 3 = − , and the upper arm be modeled by c 4, 1, 2, = 〈 − − 〉 where units are in feet. (a) Find a vector d that represents the position of the hand. (b) Determine the distance of the hand from the origin. 68. The Sphere In space, the collection of all points that are the same distance from some fixed point is called a sphere. See the illustration. The constant distance is called the radius, and the fixed point is the center of the sphere. Show that an equation of a sphere with center at x y z , , 0 0 0 ( ) and radius r is x x y y z z r 0 2 0 2 0 2 2 ( ) ( ) ( ) − + − + − = [Hint: Use the Distance Formula (1).] y x z P 5 (x, y, z) P0 5 (x0, y0, z0) r In Problems 69 and 70, find an equation of a sphere with radius r and center P .0 69. r P 1; 3, 1, 1 0 ( ) = = 70. r P 2; 1, 2, 2 0 ( ) = = In Problems 71–76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] 71. x y z x y 2 2 2 2 2 2 + + + − = 72. x y z x z 2 2 1 2 2 2 + + + − = − 73. x y z x y z 4 4 2 0 2 2 2 + + − + + = 74. x y z x4 0 2 2 2 + + − = 75. x y z x z 2 2 2 8 4 1 2 2 2 + + − + = − 76. x y z x y 3 3 3 6 6 3 2 2 2 + + + − = The work W done by a constant force F in moving an object from a point A in space to a point B in space is defined as W AB F . = ⋅ Use this definition in Problems 77–79. 77. Work Find the work done by a force of 3 newtons acting in the direction i j k 2 2 + + in moving an object 2 meters from 0, 0, 0 ( ) to 0, 2, 0 . ( ) 78. Work Find the work done by a force of 1 newton acting in the direction i j k 2 2 + + in moving an object 3 meters from 0, 0, 0 ( ) to 1, 2, 2 . ( ) 79. Work Find the work done in moving an object along a vector u i j k 3 2 5 = + − if the applied force is F i j k 2 . = − − Use meters for distance and newtons for force. c b a ‘Are You Prepared?’ Answer 1. x x y y 2 1 2 2 1 2 ( ) ( ) − + − Problems 80–88 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 80. Solve: x 3 2 5 − ≥ 81. Given f x x2 3 ( ) = − and g x x x 1 2 ( ) = + − , find f g x . ( )( ) 82. Find the exact value of sin80 cos50 cos80 sin 50 . ° ° − ° ° 83. Solve the triangle. B A c 3 6 84. Find the distance between the points P 1, 2 1 ( ) = − − and P 9, 3 . 2 ( ) = 85. Form a polynomial function with real coefficients having degree 4 and zeros −i and − i 1 3 . 86. The function f x x 5 8 ( ) = − is one-to-one. Find its inverse. 87. Find the average rate of change of f x x 3tan 2 ( ) ( ) = from 8 π − to 8 . π 88. Find the area of the region enclosed by f x x 36 2 ( ) = − and g x x 6 . ( ) = −
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