Talk:Extra:Intersecting planes

Rotate $\mathbf {v} _{2b}$ about $\mathbf {n} _{2}$

In general, rotation of a vector $\scriptstyle \mathbf {v} _{\mathrm {start} }={\begin{bmatrix}x&y&z\end{bmatrix}}$about an arbitrary unit-vector $\scriptstyle \mathbf {n} ={\begin{bmatrix}u&v&w\end{bmatrix}}$ gives (1, 2):

$\mathbf {v} _{\mathrm {end} }={\begin{bmatrix}u(ux+vy+wz)(1-\cos \theta )+x\cos \theta +(-wy+vz)\sin \theta \\v(ux+vy+wz)(1-\cos \theta )+y\cos \theta +(wx-uz)\sin \theta \\w(ux+vy+wz)(1-\cos \theta )+z\cos \theta +(-vx+uy)\sin \theta \end{bmatrix}}$

In this particular case, we thus expect:

${\begin{alignedat}{2}\mathbf {v} _{2}&={\begin{bmatrix}(-wy+vz)\sin \phi \\v(vy+wz)(1-\cos \phi )+y\cos \phi \\w(vy+wz)(1-\cos \phi )+z\cos \phi \end{bmatrix}}\\&={\begin{bmatrix}(-\cos \alpha q\cos \alpha +-\sin \alpha q\sin \alpha )\sin \phi \\-\sin \alpha (-\sin \alpha q\cos \alpha +\cos \alpha q\sin \alpha )(1-\cos \phi )+q\cos \alpha \cos \phi \\\cos \alpha (-\sin \alpha q\cos \alpha +\cos \alpha q\sin \alpha )(1-\cos \phi )+q\sin \alpha \cos \phi \end{bmatrix}}\\&=q{\begin{bmatrix}-(\cos ^{2}\alpha +\sin ^{2}\alpha )\sin \phi \\\sin ^{2}\alpha (\cos \alpha -\cos \alpha )(1-\cos \phi )+\cos \alpha \cos \phi \\\cos ^{2}\alpha (-\sin \alpha +\sin \alpha )(1-\cos \phi )+\sin \alpha \cos \phi \end{bmatrix}}\\&=q{\begin{bmatrix}-\sin \phi \\\cos \alpha \cos \phi \\\sin \alpha \cos \phi \end{bmatrix}}\end{alignedat}}$