814 CHAPTER 8 Applications of Trigonometry With the unit vectors i and j, we can express any other vector 8a, b9 in the form ai + bj, as shown in Figure 42(b) on the previous page, where 83, 49 = 3i + 4j. The vector operations previously given can be restated, using ai + bj notation. i, j Form for Vectors If v = 8a, b9, then v = ai + bj, where i = 81, 09 and j = 80, 19. The Dot Product and the Angle between Vectors The dot product of two vectors is a real number, not a vector. It is also known as the inner product. Dot products are used to determine the angle between two vectors, to derive geometric theorems, and to solve physics problems. Dot Product The dot product of the two vectors u = 8a, b9 and v = 8c, d9 is denoted u # v, read “u dot v,” and given by the following. u # v =ac +bd That is, the dot product of two vectors is the sum of the product of their first components and the product of their second components. EXAMPLE 5 Finding Dot Products Find each dot product. (a) 82, 39 # 84, -19 (b) 86, 49 # 8-2, 39 SOLUTION (a) 82, 39 # 84, -19 = 2142 + 31-12 = 5 (b) 86, 49 # 8-2, 39 = 61-22 + 4132 = 0 S Now Try Exercises 49 and 51. The following properties of dot products can be verified using the definitions presented so far. Properties of the Dot Product For all vectors u, v, and w and real numbers k, the following hold true. (a) u # v =v # u (b) u # 1v +w2 =u # v +u # w (c) 1u +v2 # w=u # w+v # w (d) 1ku2 # v =k1u # v2 =u # 1kv2 (e) 0 # u =0 (f) u # u = ∣ u∣2
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