College Trigonometry, 2011a

1036 Applications of Trigonometry
We prove Theorem 11.23 in cases. If θ =0,then⃗v and ⃗w have the same direction. It follows 1 that
there is a real number k>0sothat ⃗w = k⃗v. Hence, ⃗v · ⃗w = ⃗v · (k⃗v) =k(⃗v · ⃗v) =k‖⃗v‖ 2 = k‖⃗v‖‖⃗v‖.
Since k>0, k = |k|, sok‖⃗v‖ = |k|‖⃗v‖ = ‖k⃗v‖ by Theorem 11.20. Hence, k‖⃗v‖‖⃗v‖ = ‖⃗v‖(k‖⃗v‖) =
‖⃗v‖‖k⃗v‖ = ‖⃗v‖‖ ⃗w‖. Since cos(0) = 1, we get ⃗v · ⃗w = k‖⃗v‖‖⃗v‖ = ‖⃗v‖‖ ⃗w‖ = ‖⃗v‖‖ ⃗w‖ cos(0), proving
that the formula holds for θ =0. Ifθ = π, we repeat the argument with the difference being ⃗w = k⃗v
where k