Angle Bisector Vector/Geometric Proof 1

Proof

 * Angular Bisector Vector Diagram.png

As shown above:


 * Let $\gamma$ be the angle between $\mathbf u$ and $\mathbf v$.


 * Let $\alpha$ be the angle between $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$.


 * Let $\beta$ be the angle between $\mathbf u$ and $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$.

Note that $\norm {\mathbf u} \mathbf v$ is $\mathbf v$ multiplied by the length of $\mathbf u$.

By Vector Times Magnitude Same Length As Magnitude Times Vector the vectors $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$ have equal length.

So $\norm {\mathbf u} \mathbf v$, $\norm {\mathbf v} \mathbf u$ and $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ make an isosceles triangle.

Therefore:

But since $\mathbf v$ and $\norm {\mathbf u} \mathbf v$ are parallel, we also have:

Thus $\gamma = 2 \beta$, and the result follows.