Finding An Orthogonal Matrix

Question

Let $u = (0,1,2,2)^{T}$, $v = (-3,0,0,0)^{T}$. Find an orthogonal matrix $A$ such that $Au=v$ and $A = I-B$, where $B$ is a matrix of rank one.

I started by writing $A$ as $A = I - xy^{*}$ and using $Au = v$. But it does not seem to be working.

Answer 1

take $a = u-v = (3,1,2,2)^T$ and form the reflection matrix $R = I - 2aa^T/(a^Ta), $ where $I$ is the identity matrix. now you should have $Ru = v.$