Angle Bisector Vector/Algebraic Proof

Theorem
Let $\mathbf u$ and $\mathbf v$ be vectors of non-zero length.

Let $\left\Vert{\mathbf u}\right\Vert$ and $\left\Vert{\mathbf v}\right\Vert$ be their respective lengths.

Then $\left\Vert{\mathbf u}\right\Vert \mathbf v + \left\Vert{\mathbf v}\right\Vert \mathbf u$ is the angle bisector of $\mathbf u$ and $\mathbf v$.

Proof
Let $\mathbf a = \left\Vert{\mathbf u}\right\Vert \mathbf v + \left\Vert{\mathbf v}\right\Vert\mathbf u$.

Then:

Comparing the two expressions gives us:


 * $\cos \angle \mathbf u, \mathbf a = \cos \angle \mathbf a, \mathbf v$

Since the angle used in the dot product is always taken to be between $0$ and $\pi$ and cosine is injective on this interval (from Shape of Cosine Function), we have:


 * $\angle \mathbf u, \mathbf a = \angle \mathbf a, \mathbf v$

The result follows.