Michael Corral: Vector Calculus

1.4 Cross Product 25
Example 1.12. Volume of a parallelepiped: Let the vectors u, v, w in 3 represent
adjacent sides of a parallelepiped P, as in Figure 1.4.7. Show that the volume of P is
the scalar triple product u·(v×w).
Solution: Recall that the volume of a parallelepiped
is the area A of the base parallelogram
times the height h. By Theorem
1.13(b), the area A of the base parallelogram
is‖v×w‖. And we can see that since v×w is
perpendicular to the base parallelogram determined
by v and w, then the height h is
‖u‖ cosθ, whereθistheanglebetweenuand
v×w. By Theorem 1.6 we know that
cosθ= u·(v×w)
‖u‖‖v×w‖ . Hence,
vol(P)=Ah
=‖v×w‖ ‖u‖u·(v×w)
‖u‖‖v×w‖
= u·(v×w)
v×w
θ
u
w
h
v
Figure 1.4.7 Parallelepiped P
In Example 1.12 the height h of the parallelepiped is‖u‖ cosθ, and not−‖u‖ cosθ,
becausethevectoruisonthesamesideofthebaseparallelogram’splaneasthevector
v×w(sothatcosθ>0). Sincethevolumeisthesamenomatterwhichbaseandheight
we use, then repeating the same steps using the base determined by u and v (since w
is on the same side of that base’s plane as u×v), the volume is w·(u×v). Repeating
this with the base determined by w and u, we have the following result:
For any vectors u, v, w in 3 ,
u·(v×w)=w·(u×v)=v·(w×u) (1.12)
(Note that the equalities hold trivially if any of the vectors are 0.)
Since v×w=−w×v for any vectors v, w in 3 , then picking the wrong order for
the three adjacent sides in the scalar triple product in formula (1.12) will give you the
negativeofthevolumeoftheparallelepiped. Sotakingtheabsolutevalueofthescalar
triple product for any order of the three adjacent sides will always give the volume:
Theorem 1.15. If vectors u, v, w in 3 represent any three adjacent sides of a parallelepiped,
then the volume of the parallelepiped is|u·(v×w)|.
Another type of triple product is the vector triple product u×(v×w). The proof of
the following theorem is left as an exercise for the reader: