652 CHAPTER 9 Polar Coordinates; Vectors Finding Dot Products If v i j 2 3 = − and w i j 5 3 , = + find: (a) v w⋅ (b) w v ⋅ (c) v v ⋅ (d) w w⋅ (e) v (f) w Solution EXAMPLE 1 (a) v w 2 5 3 3 1 ( ) ⋅ = ⋅ + − = (b) w v 5 2 3 3 1 ( ) ⋅ = ⋅ + − = (c) v v 2 2 3 3 13 ( )( ) ⋅ = ⋅ + − − = (d) w w 5 5 3 3 34 ⋅ = ⋅ + ⋅ = (e) v 2 3 13 2 2 ( ) = + − = (f) w 5 3 34 2 2 = + = In Words A scalar multiple vα is a vector. A dot product u v ⋅ is a scalar (real number). Figure 68 A u v u 2 v u DEFINITION Dot Product If a b v i j 1 1 = + and a b w i j 2 2 = + are two vectors, the dot product v w⋅ is defined as a a b b v w 1 2 1 2 ⋅ = + (1) Since the dot product v w⋅ of two vectors v and w is a real number (a scalar), the dot product is sometimes referred to as the scalar product . The results obtained in Example 1 suggest some general properties of the dot product. THEOREM Properties of the Dot Product If u , v , and w are vectors, then Commutative Property u v v u ⋅ = ⋅ (2) Distributive Property u v w u v u w ( ) ⋅ + = ⋅ + ⋅ (3) v v v 2 ⋅ = (4) 0 v 0 ⋅ = (5) Proof We prove properties (2) and (4) here and leave properties (3) and (5) as exercises (see Problems 38 and 39). To prove property (2), let a b u i j 1 1 = + and a b v i j. 2 2 = + Then a a b b a a b b u v v u 1 2 1 2 2 1 2 1 ⋅ = + = + = ⋅ To prove property (4), let a b v i j. = + Then a b v v v 2 2 2 ⋅ = + = ■ 2 Find the Angle between Two Vectors One use of the dot product is to find the angle between two vectors. Let u and v be two vectors with the same initial point A . Then the vectors u , v , and u v − form a triangle. See Figure 68. The angle θ at vertex A of the triangle is the angle between the vectors u and v . We wish to find a formula for calculating the angle .θ
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