816 CHAPTER 8 Applications of Trigonometry EXAMPLE 6 Finding the Angle between Two Vectors Find the angle u between each pair of vectors. (a) u = 83, 49 and v = 82, 19 (b) u = 82, -69 and v = 86, 29 SOLUTION (a) cos u = u # v u v cos u = 83, 49 # 82, 19 83, 49 82, 19 Substitute values. cos u = 3122 + 4112 2 9 + 16 # 24 + 1 Use the definitions. cos u = 10 525 Simplify. cos u ≈0.894427191 Use a calculator. u ≈26.57° Use the inverse cosine function. (b) cos u = u # v u v cos u = 82, -69 # 86, 29 82, -69 86, 29 Substitute values. cos u = 2162 + 1-62122 2 4 + 36 # 236 + 4 Use the definitions. cos u = 0 Evaluate. The numerator is equal to 0. u = 90° cos-1 0 = 90° S Now Try Exercises 55 and 57. Geometric interpretation of the dot product Geometric interpretation of the dot product For angles U between 0° and 180°, cos U is positive, 0, or negative when U is less than, equal to, or greater than 90°, respectively. Therefore, the dot product of nonzero vectors is positive, 0, or negative according to this table. Dot Product Angle between Vectors Positive Acute 0 Right Negative Obtuse Thus, in Example 6, the vectors in part (a) form an acute angle, and those in part (b) form a right angle. If u # v = 0 for two nonzero vectors u and v, then cos u = 0 and u = 90°. Thus, u and v are perpendicular vectors, also called orthogonal vectors. See Figure 44. 0 u v x y u · v = 0 k2, –6l k6, 2l Figure 44 Orthogonal vectors
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