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Parallelepiped


Parallelepiped

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

V_(parallelepiped)=|A·(BxC)|
(1)
=|C·(AxB)|
(2)
=|B·(CxA)|.
(3)

In n dimensions, a parallelepiped is the polytope spanned by n vectors v_1, ..., v_n in a vector space over the reals,

 span(v_1,...,v_n)=t_1v_1+...+t_nv_n,
(4)

where t_i in [0,1] for i=1, ..., n. In the usual interpretation, the vector space is taken as Euclidean space, and the content of this parallelepiped is given by

 abs(det(v_1,...,v_n)),
(5)

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

Given k vectors v_1, ..., v_k in n-dimensional space, their convex hull (along with the zero vector)

 {sum_(i)t_iv_i:0<=t_i<=1}
(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

 A=(v_1...v_k)
(7)

is a square matrix, and the volume of the parallelepiped is given by |detA|, where the columns of A are given by the vectors v_i. More generally, a parallelepiped has k dimensional volume given by |detA^(T)A|^(1/2).

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


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

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