About the transitivility of the action of $SO(2)$

Question

Let $E^2$ be a two dimensional real vector-space and $\langle\cdot,\cdot\rangle$ an inner product on $E^2$. If $\mu:SO(2)\times E^2\to E^2$ is the action of the lie group $SO(2)$ over $E^2$, can we deduce the following statement?

Given $u,v\in E^2$, $||u||=||v||$ there exists $A\in SO(2)$ such that $\mu_A(u)=v$

Answer 1

Yes, just take the rotation matrix for angle $\varphi=\mbox{arg}(v)-\mbox{arg}(u)$.