A First Course in Linear Algebra, 2017a

184 R n
4.9.1 The Box Product
Recall that we can use the cross product to find the the area of a parallelogram. It follows that we can use
the cross product together with the dot product to find the volume of a parallelepiped.
We begin with a definition.
Definition 4.54: Parallelepiped
A parallelepiped determined by the three vectors, ⃗u,⃗v,and⃗w consists of
{r⃗u + s⃗v +t⃗w : r,s,t ∈ [0,1]}
That is, if you pick three numbers, r,s, and t each in [0,1] and form r⃗u + s⃗v +t⃗w then the collection
of all such points makes up the parallelepiped determined by these three vectors.
The following is an example of a parallelepiped.
⃗u ×⃗v
θ
⃗w
⃗v
⃗u
Notice that the base of the parallelepiped is the parallelogram determined by the vectors ⃗u and ⃗v.
Therefore, its area is equal to ‖⃗u ×⃗v‖. The height of the parallelepiped is ‖⃗w‖cosθ where θ is the angle
shown in the picture between ⃗w and ⃗u ×⃗v. The volume of this parallelepiped is the area of the base times
the height which is just
‖⃗u ×⃗v‖‖⃗w‖cosθ =(⃗u ×⃗v) •⃗w
This expression is known as the box product and is sometimes written as [⃗u,⃗v,⃗w]. You should consider
what happens if you interchange the ⃗v with the ⃗w or the ⃗u with the ⃗w. You can see geometrically from
drawing pictures that this merely introduces a minus sign. In any case the box product of three vectors
always equals either the volume of the parallelepiped determined by the three vectors or else −1 times this
volume.
Proposition 4.55: The Box Product
Let ⃗u,⃗v,⃗w be three vectors in R n that define a parallelepiped. Then the volume of the parallelepiped
is the absolute value of the box product, given by
|(⃗u ×⃗v) •⃗w|
Consider an example of this concept.