Scalar Triple Product

The Vector product

$$[{\bf A},{\bf B},{\bf C}]\equiv {\bf A}\cdot ({\bf B}\times {... ..._2 & A_3\cr B_1 & B_2 & B_3\cr C_1 & C_2 & C_3\cr}\right\vert,$$

which yields a Scalar (actually, a Pseudoscalar).

The Volume of a Parallelepiped whose sides are given by the vectors ${\bf A}$, ${\bf B}$, and ${\bf C}$ is

$$V_{\rm parallelepiped} = \vert{\bf A}\cdot ({\bf B}\times {\bf C})\vert.$$

See also Cross Product, Dot Product, Parallelepiped, Vector Triple Product

References

Arfken, G. ``Triple Scalar Product, Triple Vector Product.'' §1.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 26-33, 1985.