Is it possible to create one single matrix which contains all of these three rotation matrices?
$$R(x) = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha & 0 \\ 0 & \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \qquad R(y) = \begin{pmatrix}\cos\beta & 0 & -\sin\beta & 0\\ 0 & 1 & 0 & 0\\ \sin\beta & 0 & \cos\beta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \\R(z) = \begin{pmatrix}\cos\gamma & -\sin\gamma & 0& 0 \\ \sin\gamma & \cos\gamma & 0& 0 \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$
If yes, how would this one (4×4-)matrix look like (written out)?
Best Answer
As mentioned earlier, you just need to multiply these three matrices:
$$ R = R(\alpha) \cdot R(\beta) \cdot R(\gamma) = \begin{pmatrix}\cos\beta\cdot\cos\gamma & \cos\beta\cdot\sin\gamma & -\sin\beta & 0 \\ \sin\alpha\sin\beta\cos\gamma - \cos\alpha\sin\gamma & \sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\gamma & \sin\alpha\cos\beta & 0 \\ \cos\alpha\sin\beta\cos\gamma + \sin\alpha\sin\gamma & \cos\alpha\sin\beta\sin\gamma -\sin\alpha\cos\gamma & \cos\alpha\cos\beta & 0 \\ 0 &0 &0 &1\end{pmatrix}$$