Scalar Triple Product

The scalar triple product of three vectors , , and is denoted and defined by

			(1)
			(2)
			(3)
			(4)
			(5)

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

			(7)
			(8)
			(9)

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)

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