I have the two vectors $ V1 = \begin{pmatrix} 0.577 \\ 0.577 \\ 0.577 \end{pmatrix} $ and $ V2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $
I need to find the rotation angles when rotating from V1 to V2 using Euler Angles - I must use the rotation matrix here:
R= $\begin{bmatrix} cos(\alpha)cos(\beta) & cos(\alpha)sin(\beta)sin(\gamma)-sin(\alpha)cos(\gamma) & cos(\alpha)sin(\beta)cos(\gamma)+sin(\alpha)sin(\gamma) \\ sin(\alpha)cos(\beta) & sin(\alpha)sin(\beta)sin(\gamma)+cos(\alpha)cos(\gamma) & sin(\alpha)sin(\beta)cos(\gamma)-cos(\alpha)sin(\gamma) \\ -sin(\beta) & cos(\beta)sin(\gamma) & cos(\beta)cos(\gamma) \end{bmatrix}$
I know how to find the angles given R eg. $ \alpha = Atan2(R_{23},R_{33}) $
So what i am essentially missing is solving the equation Ax=b, where i have x and b.
I know this will yield multiple solutions, i just need any.
Any help would be greatly appriciated.
Like you mentioned, there can be many solutions. So one way is to find an axis and get the rotation angle around that. Suppose $V_1$ is not parallel to $V_2$. Then $V_1+V_2$ is along the bisector of the angle between them. Use this as rotation axis. The rotation of $V_1$ around this axis will describe a cone. When you rotate $180^\circ$ you get a vector along $V_2$. Now you can use the this formula to get the rotation matrix. Similarly, you can get an axis perpendicular to $V_1$ and $V_2$ by using the cross product. The angle of rotation is given by the scalar (dot) product of the vectors.
In case $V_1$ and $V_2$ are parallel, you can choose any vector in the perpendicular plane and rotate $180^\circ$.