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
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In dimensions, a parallelepiped is the polytope spanned by vectors , ..., in a vector space over the reals,
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where for , ..., . In the usual interpretation, the vector space is taken as Euclidean space, and the content of this parallelepiped is given by
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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)
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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
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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 .