**As the name suggests, a scalar triple product involves the (scalar) product of three vectors. How may such a product be defined?**

**Consider three vectors and . Consider the quantity . Since is a scalar, you cannot define its dot product with another vector. Thus, is a meaningless quantity.**

**However, consider the expression . Since is a vector, its dot product with is defined. Thus, is defined and is termed the scalar triple product of and This product is represented concisely as .**

**An alert reader might have noticed that another valid triple product is possible: . This is the vector triple product and is considered in the next section.**

Let us try to assign a geometrical interpretation to the scalar triple product .

First of all, make co-initial. Assume for the moment that are non-coplanar. Complete the parallelopiped with as adjacent edges:

Consider . This is a vector perpendicular to the plane containing and . We have represented it by . Let the angle between and be . What can i.e. represent ? represents the area of the parallelogram .

Thus,

(height of parallelopiped h) (Area of the base parallelogram) | |

Volume of the parallelopiped |

The therefore represents the volume of the parallelopiped with as adjacent edges.

Note that the volume of the parallelopiped could equally well have been specified as

Thus, we come to an important property of the :

that is, if the vectors are cyclically permuted, the value of the remains the same. However, note that