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Grade: 12
        how can we product three vectors
7 years ago

Answers : (1)

yours katarnak Suresh
43 Points
							

 

Scalar Triple Product

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The scalar triple product of three vectors A, B, and C is denoted [A,B,C] and defined by

[A,B,C] = A·(BxC)
(1)
= B·(CxA)
(2)
= C·(AxB)
(3)
= det(ABC)
(4)
= |A_1 A_2 A_3; B_1 B_2 B_3; C_1 C_2 C_3|
(5)

where A·B denotes a dot product, AxB denotes a cross product, det(A)=|A| denotes a determinant, and A_i, B_i, and C_i are components of the vectors A, B, and C, 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 epsilon_(ijk) as

 A·(BxC)=epsilon_(ijk)A^iB^jC^k,
(6)

where Einstein summation has been used to sum over repeated indices.

Additional identities involving the scalar triple product are

A·(BxC) = B·(CxA)=C·(AxB)
(7)
[A,B,C]D = [D,B,C]A+[A,D,C]B+[A,B,D]C
(8)
[q,q^'',q^('''')][r,r^'',r^('''')] = |q·r q·r^'' q·r^(''''); q^''·r q^''·r^'' q^''·r^(''''); q^('''')·r q^('''')·r^'' q^('''')·r^('''')|.
(9)

The volume of a parallelepiped whose sides are given by the vectors A, B, and C is given by the absolute value of the scalar triple product

 V_(parallelepiped)=|A·(BxC)|.
(10)

SEE ALSO: Cross Product, Dot Product, Parallelepiped, Vector Multiplication, Vector Triple Product

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

 

 

In mathematics, the triple product is a product of three vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.

6 years ago
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