Question

why is the internal angle bisector of 2 vectors direction sum of the vectors direction

Answer 1

The internal angle bisector of two vectors is in the direction of the sum of their unit vectors because the sum of the unit vectors forms equal angles with each of the original vectors.

Answer 2

Let $\vec{a}$ and $\vec{b}$ be two vectors, and let $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$ and $\hat{b} = \frac{\vec{b}}{|\vec{b}|}$ be their respective unit vectors. The vector $\vec{d} = \hat{a} + \hat{b}$ bisects the angle between $\vec{a}$ and $\vec{b}$ because it makes equal angles with both $\vec{a}$ and $\vec{b}$. This can be shown by calculating the cosine of the angles between $\vec{d}$ and $\vec{a}$, and between $\vec{d}$ and $\vec{b}$, which are found to be equal.