Q:

Is rotation in $\mathbb R^3$ determined by action on a unit vector?

Say $X, Y$ belong to SO$(3)$, so that they are counterclockwise rotations in space. Let $\vec v :=(0,0,1)$. 

If $X\vec v=Y\vec v$, then does it follows that $X=Y$?

Based on my weak geometric intuition, I'm not sure if this is true. Besides a geometric explanation of the answer, is there a nice proof (one way or the other) using the algebraic properties of orthogonal matrixes with determinant one?

A:

No: imagine rotating by any angle about the axis which points in the direction of the vector $v$.  Every such rotation fixes $v$.  Explicitly, these are the rotations given by matrices of the form $$\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{pmatrix}.$$

A:

No: pick two different rotations in $\mathbb{R}^2$ and take the corresponding elements of $SO(3)$ that rotate the $xy$ plane and fix $v$.

