Uniqueness of the unitary operator that transforms a vector $\vec{a}$ to $\vec{b}$

Question

Suppose $\vec{a}$ and $\vec{b}$ are $n\times 1$ complex column vectors.

How can I find the $n\times n$ unitary matrix that transforms $\vec{a}$ to $\vec{b}$?

i.e., $\vec{b}=U\vec{a}$.

Furthermore, would the matrix $U$ be unique?