How to represent a vector in terms of another vector?

Question

If $u, v$ be two unit vectors. $\textbf{Then how to represent $u$ in terms of $v$? }$

I can find a matrix, $R$ such that $u=Rv$, from trial and error. How to derive the matrix analytically?

Unit vectors essentially represent direction. So $\cos(x)= u.v $ should be giving the angle between $u$ and $v$.

P.S: I am looking for the relationship between the eigenvectors of two different matrices. I know that a scalar multiplication of one vector will not give another vector.

Answer 1

If $u,v$ are unit vectors, the matrix $R=uv^T$ does the trick: $$ Rv = uv^Tv = u. $$