triple product

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There are two ways to multiply three vectors \mathbf{a}\,, \mathbf{b}\, and \mathbf{c}\, in \mathbb{R}^3\,: the scalar triple product, and the vector triple product.

Scalar triple product

The scalar triple product yields the volume of the parallelepiped spanned by \mathbf{a}\,, \mathbf{b}\, and \mathbf{c}\,, and is equal to

\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})= \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})= \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b}) .

In components,

\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) = \epsilon_{ijk} a^i b^j c^k\,,

where \epsilon_{ijk}\, is the Levi-Civita symbol and summation is implied. This can also be written in terms of the determinant of the matrix with columns (or rows) equal to \mathbf{a}\,, \mathbf{b}\, and \mathbf{c}\,:

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

Vector triple product

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

In components,

[\mathbf{a} \times (\mathbf{b} \times \mathbf{c})]_i\, = \epsilon_{ijk} \epsilon_{klm} a^j b^l c^m\,,
= (\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}) a^j b^l c^m\,,
= (a_j c^j) b_i- (a_j b^j) c_i\,.
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