There are a number of algebraic identities involving sets of four vectors. An identity known as Lagrange's identity is
given by
 |
(1)
|
(Bronshtein and Semendyayev 2004, p. 185).
Letting 
,
a number of other useful identities include
where ![[A,B,C]](/images/equations/VectorQuadrupleProduct/Inline17.svg)
denotes the scalar triple product. Equation
(◇) turns out to be relevant in the computation of the point-line
distance in three dimensions.
See also
Lagrange's Identity,
Scalar Triple Product,
Vector Multiplication,
Vector Triple Product
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References
Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, pp. 18-20,
1989.Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and
Muehlig, H. Handbook
of Mathematics, 4th ed. New York: Springer-Verlag, 2004.Griffiths,
D. J. Introduction
to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, p. 13, 1981.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 114, 1953.Referenced
on Wolfram|Alpha
Vector Quadruple Product
Cite this as:
Weisstein, Eric W. "Vector Quadruple Product."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorQuadrupleProduct.html
Subject classifications
![[A,B,D]C-[A,B,C]D](/images/equations/VectorQuadrupleProduct/Inline13.svg)
![[C,D,A]B-[C,D,B]A,](/images/equations/VectorQuadrupleProduct/Inline16.svg)
