The algebraic identity
(1)
Letting Lagrange's identity .
The
(2)
The
(3)
(Morse and Feshbach 1953, p. 114; Griffiths 1981, p. 13; Arfken 1985, p. 32), where dot product and cross product . Note
that this identity itself is sometimes known as Lagrange's
identity (Bronshtein and Semendyayev 2004, p. 185).
See also Lagrange's Identity
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References Arfken, G. "Triple Scalar Product, Triple Vector Product." §1.5 in Mathematical
Methods for Physicists, 3rd ed. Bronshtein,
I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 4th ed. Griffiths,
D. J. Introduction
to Electrodynamics. Mitrinović,
D. S. Analytic
Inequalities. Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. Referenced
on Wolfram|Alpha Binet-Cauchy Identity
Cite this as:
Weisstein, Eric W. "Binet-Cauchy Identity."
From MathWorld https://mathworld.wolfram.com/Binet-CauchyIdentity.html
Subject classifications
and 
gives Lagrange's identity.
case gives
case is equivalent to the vector identity
is the dot product and 
is the cross product. Note
that this identity itself is sometimes known as Lagrange's
identity (Bronshtein and Semendyayev 2004, p. 185).
