SECTION 9.6 Vectors in Space 659 42. Show that the projection of v onto i is v i i. ( ) ⋅ Then show that we can always write a vector v as v v i i v j j ( ) ( ) = ⋅ + ⋅ 43. Let v and w denote two nonzero vectors. Show that the vectors w v v w + and w v v w − are orthogonal. 44. Let v and w denote two nonzero vectors. Show that the vector v wα− is orthogonal to w if v w w . 2 α = ⋅ 45. Given vectors u i j5 = + and y v i j 4 , = + find y so that the angle between the vectors is 60 .° † 46. Given vectors x u i j2 = + and v i j 7 3 , = − find x so that the angle between the vectors is 30 .° 47. Given vectors x u i j 2 3 = + and x v i j8 , = − find x so that u and v are orthogonal. 48. In the definition of work given in this section, what is the work done if F is orthogonal to AB ? 49. Challenge Problem (a) If u and v have the same magnitude, show that u v + and u v − are orthogonal. u v 2v (b) Use this to prove that an angle inscribed in a semicircle is a right angle (see the figure). 50. Challenge Problem Prove the polarization identity , u v u v u v 4 2 2 ( ) + − − = ⋅ 51. Create an application (different from any found in the text) that requires a dot product. Explaining Concepts 57. What is the function that is graphed after the graph of y x 3 = is shifted left 4 units and up 9 units? 58. Find all asymptotes of the graph of f x x x x 2 5 2 15 . 2 2 ( ) = − − − 59. Find the exact value of cos80 cos70 sin80 sin70 . ° ° − ° ° 60. Find the vertex and determine if the graph of f x x x 2 3 12 10 2 ( ) = − + is concave up or concave down. 61. If f x x 1 9 2 3 2 ( ) ( ) = + and g x x 3 tan , ( ) = show that ( )( ) = f g x x 1 27 sec3 52. Find the average rate of change of f x x x5 27 3 2 ( ) = − + from 3− to 2. 53. Find the exact value of 5 cos60 2 tan 4 . π ° + Do not use a calculator. 54. Establish the identity: 1 sin 1 tan 1 2 2 θ θ ( )( ) − + = 55. Volume of a Box An open-top box is made from a sheet of metal by cutting squares from each corner and folding up the sides. The sheet has a length of 19 inches and a width of 13 inches. If x is the length of one side of each square to be cut out, write a function, V x( ) , for the volume of the box in terms of x . 56. Solve: 7 3 2 x x 1 4 = ⋅ − + Problems 52–61 are based on previously learned material. 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 ‘Are You Prepared?’ Answers 1. c a b ab C 2 cos 2 2 2 = + − † Courtesy of the Joliet Junior College Mathematics Department 9.6 Vectors in Space Now Work the ‘Are You Prepared?’ problem on page 667. • Distance Formula (Section 1.2, pp. 13–16) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find the Distance between Two Points in Space (p. 660) 2 Find Position Vectors in Space (p. 661) 3 Perform Operations on Vectors (p. 662) 4 Find the Dot Product (p. 663) 5 Find the Angle between Two Vectors (p. 664) 6 Find the Direction Angles of a Vector (p. 664)
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