### Coordinates and vectors

Given two vectors $\vec {a}$ and $\vec {b}$, represent vector $\vec {a}$ as a sum of two vectors, $\vec {a}_\parallel$ and $\vec {a}_\perp$, $\vec {a}=\vec {a}_\parallel+\vec {a}_\perp$, such that $\vec {a}_\parallel \parallel \vec {b}$ and $\vec {a}_\perp \perp \vec {b}$.