664 CHAPTER 9 Polar Coordinates; Vectors Finding the Angle between Two Vectors Find the angle θ between u i j k 2 3 6 = − + and v i j k 2 5 . = + − Solution EXAMPLE 7 Compute the quantities u v u , , ⋅ and v . ( ) ( ) ( ) ( ) ⋅=⋅+−⋅+−=− = + − + = = = + + − = u v u v 2 2 3 5 6 1 17 2 3 6 49 7 2 5 1 30 2 2 2 2 2 2 By formula (4), if θ is the angle between u and v , then u v u v cos 17 7 30 0.443 θ = ⋅ = − ≈ − So, cos 0.443 116.3 . 1 θ ( ) ≈ − ≈ ° − The dot product in space has the same properties as the dot product in the plane. THEOREM Properties of the Dot Product If u , v , and w are vectors, then Commutative Property u v v u ⋅ = ⋅ Distributive Property u v w u v u w ( ) ⋅ + = ⋅ + ⋅ ⋅ = ⋅ = v v v 0 v 0 2 5 Find the Angle between Two Vectors The angle θ between two vectors in space follows the same formula as for two vectors in the plane. 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 cos θ = ⋅ (4) Now Work PROBLEM 51 6 Find the Direction Angles of a Vector A nonzero vector v in space can be described by specifying its magnitude and its three direction angles , , α β and .γ These direction angles are defined as x y z v i v j v k theanglebetween and , thepositive -axis, 0 theanglebetween and , thepositive -axis, 0 theanglebetween and , thepositive -axis, 0 α α π β β π γ γ π = ≤ ≤ = ≤ ≤ = ≤ ≤ See Figure 84 on the next page.
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