Parallelepiped (original) (raw)
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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
In 
dimensions, a parallelepiped is the polytope spanned by 
vectors 
, ..., 
in a vector space over the reals,
 |
(4) |
|---|
where ![t_i in [0,1]](https://mathworld.wolfram.com/images/equations/Parallelepiped/Inline17.svg)
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 
.
See also
Cuboid, Determinant, Differential _k_-Form, Exterior Algebra, Parallelogram, Prismatoid,Rhombohedron, Volume Element, Volume Integral, Zonohedron
Portions of this entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd and Weisstein, Eric W. "Parallelepiped." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parallelepiped.html