819 8.4 Algebraically Defined Vectors and the Dot Product Let u = 8-2, 19, v = 83, 49, and w= 8-5, 129. Evaluate each expression. 67. 13u2 # v 68. u # 13v2 69. u # v - u # w 70. u # 1v - w2 Determine whether each pair of vectors is orthogonal. See Example 6(b). 71. 81, 29, 8-6, 39 72. 81, 19, 81, -19 73. 81, 09, 822, 09 74. 83, 49, 86, 89 75. 25 i - 2j, -5i + 225 j 76. -4i + 3j, 8i - 6j 77. 2i + 3j, -6i + 4j 78. i + 322 j, 6i - 22 j 79. (Modeling) Measuring Rainfall Suppose that vector R models the amount of rainfall in inches and the direction it falls, and vector A models the area in square inches and the orientation of the opening of a rain gauge, as illustrated in the figure. The total volume V of water collected in the rain gauge is given by V = R # A . This formula calculates the volume of water collected even if the wind is blowing the rain in a slanted direction or the rain gauge is not exactly vertical. Let R= i - 2j and A= 0.5i + j. (a) Find R and A to the nearest tenth. Interpret the results. (b) Calculate V to the nearest tenth, and interpret this result. 80. Concept Check In Exercise 79, for the rain gauge to collect the maximum amount of water, what should be true about vectors R and A? 7 6 5 4 3 2 1 A R Relating Concepts For individual or collaborative investigation (Exercises 81 – 86) Consider the two vectors u and v shown. Assume all values are exact. Work Exercises 81 – 86 in order. 0 x y u v 3 12 u1 = 1108 u2 = 2608 NOT TO SCALE 81. Use trigonometry alone (without using vector notation) to find the magnitude and direction angle of u + v. Use the law of cosines and the law of sines in your work. 82. Find the horizontal and vertical components of u, using a calculator. 83. Find the horizontal and vertical components of v, using a calculator. 84. Find the horizontal and vertical components of u + v by adding the results obtained in Exercises 82 and 83. 85. Use a calculator to find the magnitude and direction angle of the vector u + v. 86. Compare the answers in Exercises 81 and 85. What do you notice? Which method of solution do you prefer?
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