In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let , , and be the basis vectors defining a three-dimensional parallelepiped. Then the parallelepiped has volume given by the scalar triple product

			(1)
			(2)
			(3)

In dimensions, a parallelepiped is the polytope spanned by vectors , ..., in a vector space over the reals,

(4)

where for , ..., . In the usual interpretation, the vector space is taken as Euclidean space, and the content of this parallelepiped is given by

(5)

where the sign of the determinant is taken to be the "orientation" of the "oriented volume" of the parallelepiped.

Given vectors , ..., in -dimensional space, their convex hull (along with the zero vector)

(6)

is called a parallelepiped, generalizing the notion of a parallelogram, or rather its interior, in the plane. If the number of vectors is equal to the dimension, then

(7)

is a square matrix, and the volume of the parallelepiped is given by , where the columns of are given by the vectors . More generally, a parallelepiped has dimensional volume given by .

When the vectors are tangent vectors, then the parallelepiped represents an infinitesimal -dimensional volume element. Integrating this volume can give formulas for the volumes of -dimensional objects in -dimensional space. More intrinsically, the parallelepiped corresponds to a decomposable element of the exterior algebra .

Portions of this entry contributed by Todd Rowland

CITE THIS AS:

Rowland, Todd and Weisstein, Eric W. "Parallelepiped." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parallelepiped.html