\(v'=\cos(\theta)\,v+\sin(\theta)\,n\times v + (1-\cos(\theta))\,(v\cdot n)\,n = R(n,\theta)\,v\)
\(v'= \cos(\theta) v + \sin(\theta)\,\left(\begin{array}{l}n_x \\ n_y \\ n_z\end{array}\right)\times v +
(1-\cos(\theta))\;(\begin{array}{ccc}n_x & n_y & n_z\end{array})\,v\,\left(\begin{array}{c}n_x \\ n_y \\ n_z \end{array}\right)\)
\(v'= \cos(\theta)\,v + \sin(\theta)\,\underbrace{\left(\begin{array}{ccc}0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x &
0\end{array}\right)}_{\mathrm{K}}\,v + (1-\cos(\theta))\,\underbrace{\left(\begin{array}{ccc} n_x^2 & n_x\,n_y & n_x\,n_z \\ n_x\,n_y & n_y^2 & n_y\,n_z \\
n_x\,n_z & n_y\,n_z & n_z^2 \end{array}\right)}_{\mathrm{K^2}}\,v \)
\(v'= \left( \begin{array}{ccc} \cos(\theta)+n_x^2 (1-\cos(\theta)) & n_x n_y (1-\cos(\theta))-n_z \sin(\theta) & n_x n_z
(1-\cos(\theta))+n_y \sin(\theta) \\ \\ n_x n_y (1-\cos(\theta)) + n_z \sin(\theta) & \cos(\theta)+n_y^2 (1- \cos(\theta)) & n_y n_z (1-\cos(\theta))-n_x
\sin(\theta) \\ \\ n_x n_z (1-\cos(\theta))-n_y \sin(\theta) & n_y n_z (1-\cos(\theta))+n_x \sin(\theta) & \cos(\theta) + n_z^2 (1-\cos(\theta))
\end{array} \right)\,v \)