Part 1: The Triple Scalar Product

In the last section, we learned that ||u×v|| is the area of the parallelogram spanned by u and v, and that u×v is perpendicular to both u and v.  In this section, we look at three important applications of the cross product-the volume of a parallelepiped, the equation of a plane through 3 points, and the triple vector product.

If u,v,and w share a common initial point, then the set of terminal points of the vectors su+tv+rw for 0 £ r,s,t £ 1 is called the parallelepiped spanned by u,v,and w:

Maple/javaview image

Let's calculate the volume of a parallelepiped spanned by vectors u, v, and w.

EXAMPLE 1    Find the volume of the parallelpiped spanned by u = á 2,0,0 ñ,  v = á1,3,0 ñ, and w = á 1,0,3 ñ . The figure below is drawn as if all vectors have their initial points at the origin.

Maple/javaview image

Solution: The cross product of u and v  is

u×v =  0 0 3 0 ,   0 2 0 1 ,   2 0 1 3 = á 0,0,6 ñ

The dot product with w yields the volume:

Volume = | ( u×v) ·w| = | á 1,0,3 ñ · á0,0,6 ñ | = 18

      

Because our choice of labels is arbitrary, we should get the same volume regardless of which pair we cross and which we dot.  To illustrate, consider that in example 1, we could have calculated 

v×w  =   3 0 0 3 , 0 1 3 1 , 1 3 1 0 = á 9,-3,-3 ñ

and clearly, | u·( v×w)| = | á 2,0,0 ñ · á9,-3,-3 ñ | = 18.  In fact, the calculations have the same sign, so that we actually have

( u×v) · w = u · ( v×w)

because both represent the volume of the parallelepiped. That is, the order of the cross and dot product can be interchanged, which leads to the following:       

Theorem 4.2: If u,v,w are vectors, then they satisfy the triple scalar product

u·( v×w) = ( u×v) ·w

(1)

Moreover, the magnitude of the triple scalar product is the volume of the parallelepiped spanned by u, v, and w.