Triple product

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Parallelepiped spanned by vectors A, B, and C (shown in red).

In analytic geometry, a triple product is a common term for a product of three vectors A, B, and C leading to a scalar (a number). The absolute value of this scalar is the volume V of the parallelepiped spanned by the three vectors:

where B × C is the cross product of two vectors (resulting into a vector) and the dot indicates the inner product between two vectors (a scalar).

The triple product is sometimes called the scalar triple product to distinguish it from the vector triple product A×(B×C). The scalar triple product is often written as [A B C]. The vector triple product can be expanded by the aid of the baccab formula.

Explanation

Let n be a unit normal to the parallelogram spanned by B and C (see figure). Let h be the height of the terminal point of the vector A above the base of the parallelepiped. Recall:

Volume V of parallelepiped is height h times area S of the base.

Note that h is the projection of A on n and that the area S is the length of the cross product of the vectors spanning the base,

Use

where it is used that

(The unit normal n has the direction of the cross product B × C).

If A, B, and C do not form a right-handed system, An < 0 and we must take the absolute value: | A• (B×C)|.

Triple product as determinant

Take three orthogonal unit vectors i , j, and k and write

The triple product is equal to a 3 × 3 determinant

Indeed, writing the cross product as a determinant we find

Since a determinant is invariant under cyclic permutation of its rows, it follows

Reference

M. R. Spiegel, Theory and Problems of Vector Analysis, Schaum Publishing, New York (1959) p. 26