View source for Triple product

A [[ternary operation]], such as a [[trilinear mapping]] or a [[triple system]].

In three-dimensional vector geometry the '''scalar triple product''' or ''[[mixed product]]'' of vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ is
$$
[\mathbf{a},\mathbf{b},\mathbf{c}] = \mathbf{a} \times \mathbf{b} \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{b} \times \mathbf{c}\ ;
$$
it is the (signed) volume of the parallelepiped with edges in the directions $\mathbf{a},\mathbf{b},\mathbf{c}$.  The product is unchanged by cyclic permutation of the factors, 
$$
[\mathbf{a},\mathbf{b},\mathbf{c}] = [\mathbf{b},\mathbf{c},\mathbf{a}] = [\mathbf{c},\mathbf{a},\mathbf{b}]
$$
and reversed in sign by a transposition of factors
$$
[\mathbf{a},\mathbf{b},\mathbf{c}] = - [\mathbf{b},\mathbf{a},\mathbf{c}] \ .
$$
 
The '''vector triple product''' is
$$
\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \ .
$$


====References====
* D. E. Rutherford, Vector Methods: applied to differential geometry, mechanics, and potential theory, Edinburgh (1939) {{ZBL|65.0732.03}} repr. Dover (2004) {{ISBN|0-486-43903-8}} {{ZBL|1084.26006}}