Past exam of the options I course of the University of Oxford / 2007 / P / P1 / Solution

A Givens rotation equals the identity except in coordinates $i,j$, where its block can be taken asThe two block columns are orthonormal because $c^2+s^2=1$; other coordinates are unchanged, so the full matrix is orthogonal. If $(v_i,v_j)\ne (0,0)$, take $r=\sqrt{v_i^2+v_j^2}$, $c=v_i/r$ and $s=v_j/r$. The transformed pair is $(r,0)$. If both entries are zero, the identity already suffices.

For the specified vector one possible matrix isIts rows are orthonormal. It is not unique: premultiply by any orthogonal transformation fixing the second coordinate axis pointwise. This rotates or reflects the other two output coordinates without changing the required image. Equivalently its second row must be $v^T/\Vert v\Vert$, but any orthonormal completion of that row works.