# Triple product

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, **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

## Reference

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