The scalar triple product of three vectors
where dot product , cross product ,
determinant , and pseudoscalar
(i.e., it reverses sign under inversion). The scalar triple product can also be written
in terms of the permutation symbol
(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 absolute value
of the scalar triple product
(10)
See also Cross Product ,
Dot Product ,
Parallelepiped ,
Vector
Multiplication ,
Vector Triple Product
Explore with Wolfram|Alpha
References Arfken, G. "Triple Scalar Product, Triple Vector Product." §1.5 in Mathematical
Methods for Physicists, 3rd ed. Aris, R. "Triple Scalar Product." §2.34 in Vectors,
Tensors, and the Basic Equations of Fluid Mechanics. Griffiths, D. J. Introduction
to Electrodynamics. Jeffreys,
H. and Jeffreys, B. S. "The Triple Scalar Product." §2.091 in
Methods
of Mathematical Physics, 3rd ed. Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. Referenced
on Wolfram|Alpha Scalar Triple Product
Cite this as:
Weisstein, Eric W. "Scalar Triple Product."
From MathWorld https://mathworld.wolfram.com/ScalarTripleProduct.html
Subject classifications
, 
, and 
is denoted ![[A,B,C]](/images/equations/ScalarTripleProduct/Inline4.svg)
and defined by
![[A,B,C]](/images/equations/ScalarTripleProduct/Inline5.svg)
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
![[A,B,C]D](/images/equations/ScalarTripleProduct/Inline33.svg)
![[D,B,C]A+[A,D,C]B+[A,B,D]C](/images/equations/ScalarTripleProduct/Inline35.svg)
![[q,q^',q^('')][r,r^',r^('')]](/images/equations/ScalarTripleProduct/Inline36.svg)
, 
, and 
is given by the absolute value
of the scalar triple product
