The scalar triple product of three vectors , , and is denoted 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. Referenced
on Wolfram|Alpha Scalar Triple Product
Cite this as:
Weisstein, Eric W. "Scalar Triple Product."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ScalarTripleProduct.html
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