862 CHAPTER 8 Applications of Trigonometry Concepts Examples Two forces of 25 newtons and 32 newtons act on a point in a plane. If the angle between the forces is 62°, find the magnitude of the resultant force. v 1188 62° 32 32 25 25 P S Q R The resultant force divides a parallelogram into two triangles. The measure of angle Q in the figure is 118°. We use the law of cosines to find the desired magnitude. v 2 = 252 + 322 - 212521322 cos 118° v 2 ≈2400 v ≈49 The magnitude of the resultant force is 49 newtons. Magnitude and Direction Angle of a Vector The magnitude (length) of vector u = 8a, b9 is given by the following. ∣ u∣ =!a2 +b2 The direction angle u satisfies tan u = b a , where a≠0. If u = 8a, b9 has direction angle u, then u = 8 u cos u, u sin u9. 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, b 9 −8 c, d 9 =8 a, b 9 + 1 −8 c, d 92 =8 a −c, b −d 9 i, j Form for Vectors If v = 8a, b9, then v = ai + bj, where i = 81, 09 and j = 80, 19. Vector Sum The sum of two vectors is also a vector. There are two ways to find the sum of two vectors A and B geometrically. 1. The vector with the same initial point as A and the same terminal point as B is the sum A+ B. B A A + B 2. The diagonal of the parallelogram with the same initial point as A and B is the sum A+ B. This is the parallelogram rule. B A A + B Find the magnitude and direction angle of vector u in the figure. u = 3A223 B 2 + 22 = 216 = 4 Magnitude tan u = 2 223 = 123 # 23 23 = 23 3 , so u = 30°. For u defined above, u = 84 cos 30°, 4 sin 30°9 = 8223, 29. cos 30° = 23 2 ; sin 30° = 1 2 Find each of the following. 84, 69 + 8-8, 39 = 8-4, 99 58-2, 19 = 8-10, 59 -8-9, 69 = 89, -69 84, 69 - 8-8, 39 = 812, 39 If u = 8223, 29 as above, then u = 223 i + 2j. 8.3 Geometrically Defined Vectors and Applications 8.4 Algebraically Defined Vectors and the Dot Product x y 1 1 2 2 3 0 u u = k2Ë3, 2l
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