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
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.