The scalar triple product of three vectors 
, 
, and 
is denoted ![[A,B,C]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FScalarTripleProduct%2FInline4.svg){kind=small}
and defined by
where 
denotes a dot product, 
denotes a cross product,
denotes a determinant, and 
, 
, and 
are components of the vectors 
, 
, and 
, respectively. The scalar triple product is a pseudoscalar
(i.e., it reverses sign under inversion). The scalar triple product can also be written
in terms of the permutation symbol 
as
 |
(6) |
where Einstein summation has been used to sum over repeated indices.
Additional identities involving the scalar triple product are
The volume of a parallelepiped whose sides are given by the vectors 
, 
, and 
is given by the absolute value
of the scalar triple product
 |
(10) |
See also
Cross Product, Dot Product, Parallelepiped, Vector Multiplication, Vector Triple Product
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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.Aris, R. "Triple Scalar Product." §2.34 in Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, pp. 18-19, 1989.Griffiths, D. J. Introduction to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, p. 13, 1981.Jeffreys, H. and Jeffreys, B. S. "The Triple Scalar Product." §2.091 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 74-75, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 11, 1953.
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Cite this as:
Weisstein, Eric W. "Scalar Triple Product." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScalarTripleProduct.html