I have a question which is bothering me for days! Suppose that we have a fixed frame $XYZ$ and a moving frame $xyz$ in 3D. The moving frame is orthonormal and is defined based on the fixed one using 9 direction cosines. For instance, the unit vector $x$ is $(l_1,m_1,n_1)$ where $l_1$, $m_1$ and $n_1$ are the cosines of the angles between $x$ and $X$, $Y$ and $Z$ respectively. Similarly, we have $y=(l_2,m_2,n_2)$ and $z=(l_3,m_3,n_3)$ which are also unit vectors.
My question is: At first the moving frame $xyz$ coincides $XYZ$. Then it rotates arbitrary to form a frame with known direction cosines. How can I calculate the angle of rotation of the moving frame around its $z$ axis based on the 9 direction cosines. In other words, how much the $x$-axis rotates around the $z$-axis?
Thanks a lot for saving me!
Best Answer
I didn't read your question closely enough. Below is the answer to how to find which axis the rotation is about and the angle of rotation about that axis. I'm leaving it, though, in case it is useful.
Set this form of the rotation matrix equal to your version. Solve for the axis vector $(u_x, u_y, u_z)$ and the angle of rotation $\theta$