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A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist, [ 3 ] answering an open question of Richard Guy .
This is a list of volume formulas of basic shapes: [4]: 405–406 ... Parallelepiped – , where , , and are the ...
Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to 1 / 6 of the volume of any parallelepiped that shares three converging edges with it. The absolute value of the scalar triple product can be represented as the following absolute values of determinants:
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) In exterior algebra and geometric algebra the exterior product of two vectors is a bivector , while the exterior product of three vectors is a trivector .
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
In geometry, a rhombohedron (also called a rhombic hexahedron [1] [2] or, inaccurately, a rhomboid [a]) is a special case of a parallelepiped in which all six faces are congruent rhombi. [3] It can be used to define the rhombohedral lattice system , a honeycomb with rhombohedral cells.
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The cross product is defined by the formula [8] [9] ... Translating the above algebra into geometry, the function "volume of the parallelepiped defined by ...