Rotation Equations for 2 Angles: Combining Relationships for Easy Calculation

[Jhenrique]

Using http://www.mymathforum.com/download/file.php?id=6171 and writing the relationships:

$$\vec{\rho}\;'=R^{-1}(\phi)\vec{\rho}$$
$$\begin{bmatrix} r'\\ z'\\ \end{bmatrix} = \begin{bmatrix} cos(\phi) & sin(\phi)\\ -sin(\phi) & cos(\phi)\\ \end{bmatrix} \begin{bmatrix} r\\ z\\ \end{bmatrix}$$
and

$$\vec{r}\;'=R(\theta)\vec{r}$$
$$\begin{bmatrix} x'\\ y'\\ \end{bmatrix} = \begin{bmatrix} cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta)\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ \end{bmatrix}$$

and that $\vec{r}=\vec{x}+\vec{y}$ and $\vec{\rho}=\vec{r}+\vec{z}$.

Joinning all these relations, I ask: is possible to join theses two rotations in one unique equation?

 

[wotanub]

Indeed. Express the matrices in the same basis and matrix multiply the compose the rotations.

 

[Jhenrique]

Give me an example?