Grassman Algebra

$\textbf{a} \wedge \textbf{b} = (a_yb_z - a_zb_y)\textbf{e}_{23} + (a_zb_x - a_xb_z)\textbf{e}_{31} + (a_xb_y - a_yb_x)\textbf{e}_{12}$	Wedge product (bivector); $\textbf{e}_i \wedge \textbf{e}_j = \textbf{e}_{ij}$
$\textbf{a} \vee \textbf{b} = (a_yb_z - a_zb_y)\textbf{e}_{1} + (a_zb_x - a_xb_z)\textbf{e}_{2} + (a_xb_y + a_yb_x)\textbf{e}_{3}$	Antiwedge product
$\textbf{a} \wedge \textbf{b} = -\textbf{b} \wedge \textbf{a}$	Anti-commutative wedge property
$\textbf{a} \wedge \textbf{a} = 0$	Wedge product zero property
$\textbf{a} \wedge \textbf{b} \wedge \textbf{c} = (a_xb_yc_z + a_yb_zc_x + a_zb... ... a_yb_xc_z - a_zb_yc_x) (\textbf{e}_1 \wedge \textbf{e}_2 \wedge \textbf{e}_3)$	Triple wedge product (trivector)
$gr(\textbf{A} \wedge \textbf{B}) = gr(\textbf{A}) + gr(\textbf{B})$	Grade property (gr); A and B are k-vectors (e.g. bi or tri) where k is the grade
$gr(\textbf{A} \wedge \textbf{B}) = -1^{gr(\textbf{A})gr(\textbf{B})}(\textbf{B} \wedge \textbf{A})$	Negation commutative property
$\textbf{E}_n = \textbf{e}_{12...n} = \textbf{e}_1 \wedge \textbf{e}_2 \wedge ... \textbf{e}_n$	Unit volume element for a n-dim grassman alg
$\textbf{A} \wedge \overline{\textbf{A}} = \textbf{E}_n$	n-vector complement (contains all basis elements NOT present in original element A)
$\textbf{a} \times \textbf{b} = \overline{\textbf{a} \wedge \textbf{b}}$	Cross product and wedge product complement
$\underline{\textbf{A}} = (-1)^{k(n-k)}\overline{\textbf{A}}$	Left complement when n is even
$\overline{\textbf{A} \wedge \textbf{B}} = \overline{\textbf{A}} \vee \overline{\textbf{B}}$	Complement property (and similar for all variations, like DeMorgan's rules)
$\textbf{a} \cdot \textbf{b} = \textbf{a} \vee \overline{\textbf{b}}$	Dot product (interior product)
$\textbf{p} \wedge \textbf{q} = (q_x - p_x)\textbf{e}_{41} + (q_y - p_y)\textbf{... ...{e}_{23} + (p_zq_x - p_xq_z)\textbf{e}_{31} + (p_xq_y - p_yq_x)\textbf{e}_{12}$	Line representation from homogenous points p,q (w = 1)
$\textbf{p} \wedge \textbf{L} = (L_{vy}p_z - L_{vz}p_y + L_{mx})\overline{\textb... ...e{\textbf{e}_3} + (-L_{mx}p_x - L_{my}p_y - L_{mz}p_z)\overline{\textbf{e}_4}$	Plane rep from 3 homogeneous points p,q,r, L = $q \wedge r$ in plucker coords rep ($\{v\vert m\}$)
$\textbf{f} \vee \textbf{g} = (f_zg_y - f_yg_z)\textbf{e}_{41} + (f_xg_z - f_zg_... ...{e}_{23} + (f_yg_w - f_wg_y)\textbf{e}_{31} + (f_zg_w - f_wg_z)\textbf{e}_{12}$	plane intersection (a line) of planes f,g
$\textbf{f} \vee \textbf{L} = (L_{my}f_z - L_{mz}f_y + L_{vx}f_w)\textbf{e}_1 + ... ..._x + L_{vz}f_w)\textbf{e}_3 + (-L_{vx}f_x - L_{vy}f_y - L_{vz}f_z)\textbf{e}_4$	Intersection of plane f, line L (a point)
$d = \frac{\textbf{L}_1 \vee \textbf{L}_2}{\vert\vert \textbf{v}_1 \wedge \textbf{v}_2\vert\vert}$	Distance between lines L1,L2 ( $L = \{v \vert m\}$)
$d = \frac{\textbf{p} \vee \textbf{f}}{\vert\vert \textbf{n} \vert\vert}$	Distance between point p and plane f ( $f = \{\textbf{n} \vert d\}$)

$[\textbf{a} \textbf{b} \textbf{c}]^{-1} = \frac{1}{\underline{\textbf{a} \wedg... ... \wedge \textbf{c}} \\ \underline{\textbf{a} \wedge \textbf{b}} \end{bmatrix}$	Matrix inverse with col vecs a,b,c
$[\textbf{a} \textbf{b} \textbf{c} \textbf{d}]^{-1} = \frac{1}{\underline{\text... ... \\ -\underline{\textbf{a} \wedge \textbf{b} \wedge \textbf{c}} \end{bmatrix}$	Matrix inverse with col vecs a,b,c,d