Angle Bisector Vector/Geometric Proof 2

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Theorem

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

Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective lengths.


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


Proof

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

Thus $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the diagonal of a rhombus.


The result follows from Diagonals of Rhombus Bisect Angles.

$\blacksquare$