Formula |
Commutator |
Description |
Illustration
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$$\begin{split}\mathbf p \wedge \mathbf q =\, &(q_xp_w - p_xq_w)\,\mathbf e_{41} + (q_yp_w - p_yq_w)\,\mathbf e_{42} + (q_zp_w - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$
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$$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$
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Line containing points $$\mathbf p$$ and $$\mathbf q$$.
Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident.
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$$\begin{split}\mathbf L \wedge \mathbf p =\, &(v_yp_z - v_zp_y + m_xp_w)\,\mathbf e_{234} \\ +\, &(v_zp_x - v_xp_z + m_yp_w)\,\mathbf e_{314} \\ +\, &(v_xp_y - v_yp_x + m_zp_w)\,\mathbf e_{124} \\ -\, &(m_xp_x + m_yp_y + m_zp_z)\,\mathbf e_{321}\end{split}$$
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$$[\mathbf L, \mathbf p]^{\Large\unicode{x27D1}}_+$$
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Plane containing line $$\mathbf L$$ and point $$\mathbf p$$.
Zero if $$\mathbf p$$ lies on the line $$\mathbf L$$.
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$$\begin{split}\mathbf f \vee \mathbf g =\, &(f_zg_y - f_yg_z)\,\mathbf e_{41} + (f_xg_z - f_zg_x)\,\mathbf e_{42} + (f_yg_x - f_xg_y)\,\mathbf e_{43} \\ +\, &(f_xg_w - g_xf_w)\,\mathbf e_{23} + (f_yg_w - g_yf_w)\,\mathbf e_{31} + (f_zg_w - g_zf_w)\,\mathbf e_{12}\end{split}$$
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$$[\mathbf f, \mathbf g]^{\Large\unicode{x27C7}}_-$$
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Line where planes $$\mathbf f$$ and $$\mathbf g$$ intersect.
Direction $$\mathbf v$$ is zero if $$\mathbf f$$ and $$\mathbf g$$ are parallel.
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$$\begin{split}\mathbf L \vee \mathbf f =\, &(m_yf_z - m_zf_y + v_xf_w)\,\mathbf e_1 \\ +\, &(m_zf_x - m_xf_z + v_yf_w)\,\mathbf e_2 \\ +\, &(m_xf_y - m_yf_x + v_zf_w)\,\mathbf e_3 \\ -\, &(v_xf_x + v_yf_y + v_zf_z)\,\mathbf e_4\end{split}$$
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$$[\mathbf L, \mathbf f]^{\Large\unicode{x27C7}}_+$$
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Point where line $$\mathbf L$$ intersects plane $$\mathbf f$$.
Weight $$w$$ is zero if $$\mathbf L$$ is parallel to $$\mathbf f$$.
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$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-f_xp_w \mathbf e_{41} - f_yp_w \mathbf e_{42} - f_zp_w \mathbf e_{43} \\ +\, &(f_yp_z - f_zp_y)\,\mathbf e_{23} + (f_zp_x - f_xp_z)\,\mathbf e_{31} + (f_xp_y - f_yp_x)\,\mathbf e_{12}\end{split}$$
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$$[\mathbf p, \mathbf f]^{\Large\unicode{x27C7}}_+$$
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Line perpendicular to plane $$\mathbf f$$ passing through point $$\mathbf p$$.
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$$\begin{split}\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-v_xp_w \mathbf e_{234} - v_yp_w \mathbf e_{314} - v_zp_w \mathbf e_{124} \\ +\, &(v_xp_x + v_yp_y + v_zp_z)\,\mathbf e_{321}\end{split}$$
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$$-[\mathbf p, \mathbf L]^{\Large\unicode{x27C7}}_+$$
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Plane perpendicular to line $$\mathbf L$$ containing point $$\mathbf p$$.
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$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf L =\, &(v_yf_z - v_zf_y)\,\mathbf e_{234} + (v_zf_x - v_xf_z)\,\mathbf e_{314} + (v_xf_y - v_yf_x)\,\mathbf e_{124} \\ -\, &(m_xf_x + m_yf_y + m_zf_z)\,\mathbf e_{321}\end{split}$$
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$$[\mathbf L, \mathbf f]^{\Large\unicode{x27C7}}_-$$
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Plane perpendicular to plane $$\mathbf f$$ containing line $$\mathbf L$$.
Normal direction is $$(0,0,0)$$ if $$\mathbf L$$ is perpendicular to $$\mathbf f$$.
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