How to find a suitable orthogonal matrix Assume $x,y \in \text{R}^n$ are two vectors of the same length, how to prove that there is an orthogonal matrix $A$ such that $Ax=y$?

Thanks for your help.
You can explicitly use a Householder matrix. Normalise $x$ and $y$ so that they become unit vectors (the special case $x=y$ is trivial and is ignored here). Let $u = x-y$ and $Q = I - \frac{uu^\top}{x^\top u}$. Then $Q$ is a real symmetric and orthogonal matrix such that $Qx=y$.
Since every orthogonal matrix $A$ always has inverse, $$ AA^{t} = A^{t}A = I $$, so that if $A$ is orthogonal then any equation $Ax=y$ has a solution $$ x = (...)y $$