quadruple product

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Given four vectors \mathbf{a}\,, \mathbf{b}\,, \mathbf{c}\, and \mathbf{d}\, in \mathbb{R}^3\,, the we can form either the scalar quadruple product, or the vector quadruple product, and these may be calculated in turn using triple products or from the properties of the Levi-Civita symbol.

Scalar quadruple product

\left(\mathbf{a}\times\mathbf{b}\right)\cdot \left(\mathbf{c}\times\mathbf{d}\right) = (\mathbf{c}\cdot\mathbf{a}) (\mathbf{b}\cdot\mathbf{d}) - (\mathbf{c}\cdot\mathbf{b}) (\mathbf{a}\cdot\mathbf{d})\,.

Vector quadruple product

\left(\mathbf{a}\times\mathbf{b}\right)\times \left(\mathbf{c}\times\mathbf{d}\right) = \det(\mathbf{c},\mathbf{d},\mathbf{a})\mathbf{b} - \det(\mathbf{c},\mathbf{d},\mathbf{b})\mathbf{a}\,,

where

\det(\mathbf{c},\mathbf{d},\mathbf{a}) = \mathbf{c}\cdot (\mathbf{d}\times\mathbf{a})\,.
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