Editing Rotation matrix (section)

====Determining the angle====
To find the angle of a rotation, once the axis of the rotation is known, select a vector {{math|'''v'''}} perpendicular to the axis. Then the angle of the rotation is the angle between {{math|'''v'''}} and  {{math|''R'''''v'''}}.

A more direct method, however, is to simply calculate the [[Trace (linear algebra)|'''trace''']]: the sum of the diagonal elements of the rotation matrix. Care should be taken to select the right sign for the angle {{mvar|θ}} to match the chosen axis:
:<math>\operatorname{tr} (R) = 1 + 2\cos\theta ,</math>

from which follows that the angle's absolute value is
:<math>|\theta| = \arccos\left(\frac{\operatorname{tr}(R) - 1}{2}\right).</math>

For the rotation axis <math>\mathbf{n}=(n_1,n_2,n_3)</math>, you can get  the correct angle<ref>{{cite arXiv|last=Kuo Kan|first=Liang|date=6 October 2018|title=Efficient conversion from rotating matrix to rotation axis and angle by extending Rodrigues' formula|eprint=1810.02999|class=cs.CG}}</ref> from

<math>\left\{\begin{matrix}
\cos \theta&=&\dfrac{\operatorname{tr}(R) - 1}{2}\\
\sin \theta&=&-\dfrac{\operatorname{tr}(K_n R)}{2}
\end{matrix}\right.
</math>

where

<math>K_n=\begin{bmatrix}
0   & -n_3 & n_2\\
n_3 & 0 & -n_1\\
-n_2 & n_1 & 0\\
\end{bmatrix}
</math>