How to find Euler's angles? I have read about Euler's angles and matrices, including $zxz,zyz$, etc . I am not obliged to use a specific rotation but rather I want to figure out what angles I need to use for alpha,theta, gamma in the specific matrix. For example, I have a vector from the centre with vertex $x,y,z$ of $(0,0,15)$, I want to rotate it to $(3,-12,15)$, what angles should I use or how can I obtain these angles ?
If let's say I choose $zyz$, I am finding difficulties finding the corresponding angles after the first $x-y$ plane rotation.
Please disregard the scaling factor and just consider the orientation. 
Look forward to hearing from you
 A: In case it is the composed rotation matrix that is of interest, the method of Euler angles is not the easiest.
Step 1: extract axis and angle
A rotation which turns $u=(0,0,15)$ in the direction of $v=(3,-12,15)$ is a rotation in the plane spanned by those vectors. This is the same as a rotation about the normal of this plane, which parallel to the cross product of the two vectors, so a normalised normal vector is $n=\frac{1}{17}(4,1,0)$.
The angle you want to rotate is the same as the angle $\alpha$ between $u$ and $v$, which can be extracted from the scalar product:
$$\alpha=\arccos\left({\frac{u\cdot v}{|u|\,|v|}}\right).$$
Step 2: construct your matrix
When you have the axis-angle representation, there are several ways to proceed to get your rotation matrix $R$. A simple one is to use the so called Rodrigues rotation formula,
$$R=I+\sin{\alpha}\;[n]_\times+(1-\cos{\alpha})[n]_\times^2,$$
where $I$ is the identity matrix and $[n]_\times$ is the cross-product matrix associated with $n$, that is
$$[n]_\times=\begin{bmatrix}0 & -n_3 & n_2 \\ n_3 & 0 & -n_1 \\ -n_2 & n_1 & 0\end{bmatrix}.$$
It is very important that $n$ is normalised.
Another elegant formula for $R$ uses the matrix exponential:
$$R=e^{\alpha\left[n\right]_\times},$$
again with $[n]_\times$ as above.
A more elementary way would use the eigendecomposition.
