How can I show that any element of SO(3) can be written in the form $ Z_\Phi X_\theta Z_\Psi $ ?
Where, $$ Z_\theta = \begin{pmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} $$
$$ X_\theta = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \\ \end{pmatrix} $$
Any hints/suggestions will be greatly appreciated ?
Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.
Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.
These matrices are sort of carrying out that procedure