### Coordinates and vectors

Given two non-parallel unit vectors $\vec {e}_1$ and $\vec {e}_2$, find another unit vector $\vec {e}_3$ such that all three are in the same plane and $\vec {e}_3$ divides the angle between the first two in the way $\hat{\vec {e}_1\vec {e}_3}= 2\hat{\vec {e}_2 \vec {e}_3}$.