674 CHAPTER 9 Polar Coordinates; Vectors 41. Find a vector orthogonal to both u and v. 42. Find a vector orthogonal to both u and w. 43. Find a vector orthogonal to both u and +i j. 44. Find a vector orthogonal to both u and +j k. In Problems 45–48, find the area of the parallelogram with one corner at P1 and adjacent sides PP 1 2 and PP . 1 3 45. ( ) ( ) ( ) = = = − P P P 0, 0, 0 , 1, 2, 3 , 2, 3, 0 1 2 3 46. ( ) ( ) ( ) = = = − P P P 0, 0, 0 , 2, 3, 1 , 2, 4, 1 1 2 3 47. ( ) ( ) ( ) = = − = − P P P 1, 2, 0 , 2, 3, 4 , 0, 2, 3 1 2 3 48. ( ) ( ) ( ) = − = − = − P P P 2, 0, 2 , 2, 1, 1 , 2, 1, 2 1 2 3 In Problems 49–52, find the area of the parallelogram with vertices P P P , , , 1 2 3 and P .4 49. ( ) ( ) ( ) = = = − P P P 1, 1, 2 , 1, 2, 3 , 2, 3, 0 , 1 2 3 ( ) = − P 2, 4, 1 4 50. ( ) ( ) ( ) = = = − P P P 2, 1, 1 , 2, 3, 1 , 2, 4, 1 , 1 2 3 ( ) = − P 2, 6, 1 4 51. ( ) ( ) ( ) = − = − = − P P P 1, 2, 1 , 4, 2, 3 , 6, 5, 2 , 1 2 3 ( ) = − P 9, 5, 0 4 52. ( ) ( ) ( ) = − = − = − − P P P 1, 1, 1 , 1, 2, 2 , 3, 4, 5 , 1 2 3 ( ) = − − P 3, 5, 4 4 58. Show that if u and v are orthogonal, then × = u v u v 59. Show that if u and v are orthogonal unit vectors, then ×u v is also a unit vector. 60. Prove property (3). 61. Prove property (5). 62. Prove property (9). [Hint: Use the result of Problem 57 and the fact that if θ is the angle between u and v, then θ ⋅ = u v u v cos .] 63. Challenge Problem Show that the vector ×v w 2 3 is orthogonal to both v and w. 64. Challenge Problem If v, w, u and a b c , , are vectors for which ⋅ = ⋅ = ⋅ = a v b w c u 1 and ⋅ =⋅=⋅=⋅=⋅=⋅ = aw au bv bu cv cw 0 show that ( ) ( ) ( ) = × ⋅ × = × ⋅ × = × ⋅ × a w u v w u b u v w u v c v w u v w Applications and Extensions 53. Find a unit vector normal to the plane containing = + − v i j k 3 2 and = − + + w i j k 2 3 . 54. Find a unit vector normal to the plane containing = + − v i j k 23 and = − − − w i j k 2 4 3 . 55. Volume of a Parallelepiped A parallelepiped is a prism whose faces are all parallelograms. Let A B , , and C be the vectors that define the parallelepiped shown in the figure. The volume V of the parallelepiped is given by the formula ( ) = × ⋅ A B C V . C B A Find the volume of a parallelepiped if the defining vectors are = − + = + − A i j k B i j k 3 2 4 , 2 2, and = − − C i j k 3 6 2 . 56. Volume of a Parallelepiped Refer to Problem 55. Find the volume of a parallelepiped whose defining vectors are = + = + − A i k B i j k 6 , 2 3 8 , and = − + C i j k 8 5 6 . 57. Prove for vectors u and v that ( ) × = − ⋅ u v u v u v 2 2 2 2 [Hint: Proceed as in the proof of property (4), computing first the left side and then the right side.] Explaining Concepts 65. If ⋅ = u v 0 and × = u v 0, what, if anything, can you conclude about u and v? 69. Use properties of logarithms to write x z log4 3 as a sum or difference of logarithms. Express powers as factors. 70. Find the exact value of − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − sec sin 3 2 . 1 66. Find the exact value of ( ) − cos 1 2 . 1 67. Find two pairs of polar coordinates θ ( ) r, , one with > r 0 and the other with < r 0, for the point with rectangular coordinates ( ) − − 8, 15. Express θ in radians. 68. For = + − f x( ) 7 5, x 1 find ( ) −f x . 1 Retain Your Knowledge Problems 66–75 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.
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