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