Parallelepiped

A parallelepiped in three dimensions 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 generated by linearly independent 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 linearly independent vectors , ..., in -dimensional Euclidean space, the convex hull of all their subset sums, including the zero vector,

(6)

is called a parallelepiped, generalizing the notion of a parallelogram, including its boundary, in the plane. Let

(7)

where the columns of are the vectors . If , then is a square matrix, and the volume of the parallelepiped is . More generally, its -dimensional volume is .

When the vectors form a basis of a full-rank point lattice , the corresponding half-open set with is a fundamental parallelepiped of , and its volume is the covolume of .

When the vectors are manifold 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 Euclidean space. More intrinsically, the parallelepiped corresponds to a decomposable element of the exterior algebra .