How to show existence of an orthogonal map?

Question

I want to show that the following holds: Let $x,y\in \mathbb{R}^n\setminus\{0\}$ be given and such that $\|x\|=\|y\|$. There is an orthogonal map $T$ such that $Ty=x$ (a rotation).

How could one justify the latter statement? Probably, this is really easy but at the moment I am stuck on this one. Any (small) hint for a starting point is welcome.

Edit: I don't want to calculate the map directly by using polar coordinates unless there is no other method. I'm looking for a more abstract argument

Answer 1

There are 3 cases:

1. $y=x$

2. $y=-x$

3. $x$ and $y$ linearly independents.

The first and second cases are trivials. ($Id$ and $-Id$).

To last case you can complete $x$ and $y$ to a orthonormal basis and choose a basis change transformation such that $Tx=y$, remember that such transformations are orthornormals.