813 8.4 Algebraically Defined Vectors and the Dot Product Vector Operations Let a, b, c, d, and k represent real numbers. 8 a, b 9 +8 c, d 9 =8 a +c, b +d 9 k # 8 a, b 9 =8 ka, kb 9 If u =8 a1 , a2 9 , then −u =8−a1 , −a2 9 . 8 a, b9 −8 c, d 9 =8 a, b9 + 1 −8 c, d 92 =8 a −c, b −d 9 y x u v 0 k–2, 1l k4, 3l Figure 40 EXAMPLE 4 Performing Vector Operations Let u = 8-2, 19 and v = 84, 39. See Figure 40. Find and illustrate each of the following. (a) u + v (b) -2u (c) 3u - 2v SOLUTION See Figure 41. Figure 41 y x u u + v v 0 k–2, 1l k4, 3l k2, 4l (a) y x u –2u 0 k–2, 1l k4, –2l (b) 3u –2v 2v 3u – 2v 0 k–6, 3l k8, 6l k–8, –6l k–14, –3l y x (c) (a) u + v = 8-2, 19 + 84, 39 = 8-2 + 4, 1 + 39 = 82, 49 (b) -2u = -2 # 8-2, 19 = 8-21-22, -21129 = 84, -29 (c) 3u - 2v = 3 # 8-2, 19 - 2 # 84, 39 = 8-6, 39 - 88, 69 = 8-6 - 8, 3 - 69 = 8-14, -39 S Now Try Exercises 33, 35, and 37. A unit vector is a vector that has magnitude 1. Two very important unit vectors are defined as follows and shown in Figure 42(a). i = 81, 09 j = 80, 19 0 i j x y 0 (3, 4) 3i u = 3i + 4j u 4j x y (b) Figure 42 (a)
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