Talk:Extra:Intersecting planes

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

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

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 \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} -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{alignat} }

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

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

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 \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} }