Talk:Extra:Intersecting planes

Rotate $\mathbf {v} _{2b}$ about $\mathbf {n} _{1}$

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:

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

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }