Triple product

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This article is about mathematics. See Lawson criterion for the use of the term triple product in relation to nuclear fusion.

In vector calculus, there are two ways of multiplying three vectors together, to make a triple product.

[edit] Scalar triple product

The scalar triple product is defined as the dot product of one of the vectors with the cross product of the other two. It is a pseudoscalar: under a reflection in a plane, it flips sign.

Geometrically, this product is the (signed) volume of the parallelepiped formed by the three vectors given. It can be evaluated numerically using any one of the following equivalent characterizations:

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

The parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a vector and a scalar, which is not defined.

The scalar triple product can also be understood as the determinant of the 3-by-3 matrix having the three vectors as rows (or columns, since the determinant for a transposed matrix, is the same as the original); this quantity is invariant under coordinate rotations.

Another useful property of the scalar triple product is that if it is equal to zero, then the three vectors a, b, and c are coplanar.

[edit] Vector triple product

The vector triple product is defined as the cross product of one of the vectors with the cross product of the other two. This is known as the vector triple product because it results in a vector.

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