I've been asked to find the matrix which rotates vector $\vec{V}$ by angle $\alpha$ in the x-y plane. This I understand and I've constructed the matrix:
$R_{\alpha}= \begin{bmatrix} cos\alpha & -sin\alpha & 0\\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\ \end{bmatrix}$
The next part confuses me though. I'm told to rotate my axes (not my vector) by an angle $\alpha$ around the $z$ axis. Once this has been achieved, then I have to rotate my vector by an angle $\beta$ on the x'-z' plane.
The way I'd do this is to find $R_{\alpha}^T$ where $R_\alpha$ is the matrix defined above. This new matrix rotates my axes. From here, to have the rotation on the x'-z' plane I define the next matrix as:
$R_{x'-z'} = \begin{bmatrix} -sin\alpha & 0 & cos\alpha \\ 0 & 1 & 0 \\ cos\alpha & 0 & sin\alpha \\ \end{bmatrix}$
This might look odd and as if I've mistaken where the sines and cosines go, but if we remember the convention for rotation then it's OK.
So, am I right? Is rotating my axes the same as using the transpose of $R_\alpha$ ? If yes, could you maybe tell me why so?
How should axes be rotated if I'm wrong then?
Thanks.