SECTION 9.5 The Dot Product 653 The sides of the triangle have lengths v u , , and u v , − and θ is the included angle between the sides of length v and u . The Law of Cosines (Section 8.3) can be used to find the cosine of the included angle. u v u v u v 2 cos 2 2 2 θ − = + − Now use property (4) to rewrite this equation in terms of dot products. u v u v u u v v u v 2 cosθ ( ) ( ) − ⋅ − = ⋅ + ⋅ − (6) Then use the distributive property (3) twice on the left side of (6) to obtain uv uv uuv vuv u u u v v u v v u u v v u v 2 ( ) ( ) ( ) ( ) − ⋅ − = ⋅ − − ⋅ − = ⋅ − ⋅ − ⋅ + ⋅ = ⋅ + ⋅ − ⋅ (7) ↑ Commutative Property (2) Combining equations (6) and (7) gives uu vv uv uu vv u v u v u v 2 2 cos cos θ θ ⋅+⋅− ⋅= ⋅+⋅− ⋅ = Finding the Angle θ between Two Vectors Find the angle θ between u i j 4 3 = − and v i j 2 5 . = + Solution EXAMPLE 2 Find u v u , , ⋅ and v . u v u v 4 2 3 5 7 4 3 5 2 5 29 2 2 2 2 ( ) ( ) ⋅ = ⋅ + − ⋅ = − = + − = = + = By formula (8), if θ is the angle between u and v , then u v u v cos 7 5 29 0.26 θ = ⋅ = − ≈ − Therefore, cos 0.26 105 . 1 θ ( ) ≈ − ≈ ° − See Figure 69. THEOREM Angle between Vectors If u and v are two nonzero vectors, the angle , 0 , θ θ π ≤ ≤ between u and v is determined by the formula u v u v cosθ = ⋅ (8) Now Work PROBLEM 9(a) AND (b) 3 Determine Whether Two Vectors Are Parallel Two vectors v and w are said to be parallel if there is a nonzero scalar α so that v w. α = In this case, the angle θ between v and w is 0 or .π Figure 69 105 θ ≈ ° y u 5 4i 2 3j v 5 2i 1 5j u x
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