PlanetPhysics/Direction Cosine Matrix to Axis Angle of Rotation

The angle of rotation can be found from the [[../Trace/|trace]] of the direction cosine matrix to axis angle of rotation [[../Matrix/|matrix]] $A_{11}+A_{22}+A_{33}=3cos(\alpha )+(1-cos(\alpha ))(e_1^2+e_2^2+e_3^2)$

Noting that the axis of rotation is a [[../PureState/|unit vector]] and has a length of 1 means

$e_1^2+e_2^2+e_3^2=1$

therefore

$A_{11}+A_{22}+A_{33}=1+2cos(\alpha )$

rearranging gives

$\alpha =co{s}^{-1}(\frac{1}{2}(A_{11}+A_{22}+A_{33}-1))$

Inverse cosine is a multivalued [[../Bijective/|function]] and there are 2 possible solutions for $\alpha$. Normally, the convention is to choose the principle value such that $0<\alpha <\pi$

As long as $\alpha$ is not zero, the unit vector is given by

$\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]=\left[\begin{array}{c}\frac{(A_{23}-A_{32})}{2sin(\alpha )}\\ \frac{(A_{31}-A_{13})}{2sin(\alpha )}\\ \frac{(A_{12}-A_{21})}{2sin(\alpha )}\end{array}\right]$

Above equation should be proved at some time...