TITLE: Proving $\chi$ fixes two points in the unit sphere.
QUESTION [5 upvotes]: Let $\chi:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be an orthogonal transformation such that $\det(\chi)=1$ and $\chi$ is not the identity linear transformation.  Let $S \subset \mathbb{R}^{3}$, be the unit sphere.  Then how do we prove that $\chi$ fixes only two points of $S$?
Any ideas of proceeding for the solution?

REPLY [2 votes]: An orthogonal transformation with $\det(1)$ is a rotation. The only fixed points are the one on the rotation axis. And the intersection of the rotation axis and the unit sphere are exacty two points.