Part 4: Volume of a Parallelepiped
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:
In order to find the volume of the parallelepiped, we first notice that the parallelepiped
can be "sliced" into parts that can be rearranged to form a
new parallelepiped spanned by u, v, and proj_{u×v}(w) . Click the slideshow arrows
below the image to see this slicing and rearranging in action.





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










The new parallelepiped and the old parallelepiped have the same volume, and
because proj_{u×v}( w) is
perpendicular to the base spanned by u and v, the volume of the new parallelepiped is


( Area of base) ( height) 



u×v proj_{u×v}( w)  



u×v  
w·( u×v)
( u×v) ·( u×v)

( u×v)  



u×v 
 w·( u×v) 
u×v^{2}

u×v 





Thus, the volume of the parallelepiped spanned by u, v,
and w is
Equivalently, Volume =  ( u×v) · w 
EXAMPLE 7 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.
Solution: The cross product of u and v is
The dot product with w yields the volume:
Volume =  ( u×v) ·w = 
á 1,0,3
ñ ·
á0,0,6
ñ  = 18 
