Parallelepiped spanned by vectors A
, 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.
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,
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, A•n < 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
M. R. Spiegel, Theory and Problems of Vector Analysis, Schaum Publishing, New York (1959) p. 26