I am trying to prove that $\vec{A}=(\vec{A}\cdot \vec{n})\vec{n}+(\vec{n}\times\vec{A})\times\vec{n}$ where $\vec{n}$ is a unit vector and $\times$ indicates the cross product.
I am dealing with vectors in 3-dimensions in Klepner's book on mechanics, and so I assigned $\vec{A}$ in terms of $\hat{i}$, $\hat{j}$ and $\hat{k}$ and tried to do the same with the unit vector. That made my solution hideous.
I was wondering if someone could show me how to do the problem in $3$ dimensions elegantly.