Triple product
From Freepedia
This article is about mathematics. For the use of the term triple product in relation to nuclear fusion, see Lawson criterion.
In vector calculus, there are two ways of multiplying three vectors together, to make a triple product.
One is to find the dot product of one of the vectors with the cross product of the other two. This is known as the scalar triple product even though 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.
- <math>
\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})= \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})= \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b}) </math>
It is the determinant of the 3-by-3 matrix having the three vectors as columns; this in invariant under coordinate rotations.
The other is to find 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.
- <math>\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math>
