from wikipedia the above rotation matrix has a rotation of -74 degrees. What does it mean "around the axis (−1⁄3,2⁄3,2⁄3)"? How can I determine how many degrees is rotated on X axis, Y axis and Z axis?
2026-03-27 22:59:22.1774652362
rotation matrix to axis angle
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The axis of a rotation in $\Bbb R^3$ is the line which remains unchanged by the rotation. Think of twirling an umbrella around its shaft - the shaft always stays in the same place and this is the axis of the rotation.
The angle is measured in a plane perpendicular to the axis. So, pick any vector perpendicular to the axis; find the vector which results from rotating this vector; the angle of the rotation means the angle between these vectors.
It does not really make any sense to ask for the angle of rotation "on the $x$ axis".
The easiest way to find the angle is to use the formula $$1+2\cos\theta={\rm trace}(Q)\ ,$$ where ${\rm trace}(Q)$ means the sum of the diagonal elements of $Q$. In this case $${\rm trace}(Q)=0.36+0.60+0.60\ .$$
You can find the axis from the fact that it is an eigenvector of $Q$ corresponding to eigenvalue $1$. That is, solve $(Q-I){\bf v}=\bf0$ to find the axis $\bf v$.