Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$ > Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$. I think $S^2$ means a 2-D sphere and $SO(3)$ is the usual $SO(3)$ group. I'm unsure how to prove that $SO(3)$ acts transitively. My guess is to show that like $SO(2)$, $SO(3)$ preserves distance, but I'm unsure how that can guarantee that it will act transitively on the sphere. Any ideas if I'm thinking about this right?

It suffices to show all of $S^2$ is the orbit of a single point, say $e_1$. The equation $Ae_1=x$ says that the first column of $A$ is $x\in S^2$. To create the rest of the matrix $A$, simply take any two vectors on our sphere that are orthogonal to each other and $x$. To do this, compute $e_1\times x$ and then normalize to obtain $y$ for example, then compute $x\times y$ and normalize to obtain $z$. Observe $\det[x~y~z]=\pm1$, so we can permute $y$ and $z$ as necessary to get either $A=[x~y~z]$ or $[x~z~y]$.