Difference between revisions of "Parallelepiped"
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| − | A '''parallelepiped''' is a [[prism]] that has [[parallelograms]] for its | + | A '''parallelepiped''' is a [[prism]] that has [[parallelograms]] for its faces. Similarly, a parallelepiped is a [[hexahedron]] with six parallelogram faces. Specific parallelepipeds include the [[cube]], the [[cuboid]], and any rectangular [[prism]]. |
==Specific Cases== | ==Specific Cases== | ||
| − | A parallelepiped with all rectangular faces is a | + | A parallelepiped with all rectangular faces is a cuboid, and a parallelepiped with six rhombus faces is known as a [[rhombohedron]]. In an <math>n-</math>dimensional space, a parallelepiped is sometimes referred to as an <math>n-</math>dimensional parallelepiped, or as an <math>n-</math>parallelepiped. A cube is a parallelepiped with all [[square]] faces. |
==Volume== | ==Volume== | ||
| − | The [[volume]] of a parallelepiped is the product of area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one [[vertex]] are defined as [[ | + | The [[volume]] of a parallelepiped is the product of the area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one [[vertex]] are defined as [[vector]] <math>a, b,</math> and <math>c</math> with the specific vertex as the [[origin]], then the volume of the parallelepiped is the same as the [[scalar triple product]] of the vectors, or <math>a \cdot (b \times c)</math>. Suppose that <math>\bold{a} = a_1\bold{i}+a_2\bold{j}+a_3\bold{k}</math>, <math>\bold{b} = b_1\bold{i}+b_2\bold{j}+b_3\bold{k}</math>, <math>\bold{c} = c_1\bold{i}+c_2\bold{j}+c_3\bold{k}</math>. We then have the area of the parallelepiped is <cmath>|\text{det}(\begin{bmatrix} |
| − | + | c_1 & c_2 & c_3 \\ | |
| + | a_1 & a_2 & a_3\\ | ||
| + | b_1 & b_2 & b_3 \\ | ||
| + | \end{bmatrix})|.</cmath> | ||
Latest revision as of 19:08, 17 August 2023
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A parallelepiped is a prism that has parallelograms for its faces. Similarly, a parallelepiped is a hexahedron with six parallelogram faces. Specific parallelepipeds include the cube, the cuboid, and any rectangular prism.
Specific Cases
A parallelepiped with all rectangular faces is a cuboid, and a parallelepiped with six rhombus faces is known as a rhombohedron. In an 
dimensional space, a parallelepiped is sometimes referred to as an dimensional parallelepiped, or as an parallelepiped. A cube is a parallelepiped with all square faces.
Volume
The volume of a parallelepiped is the product of the area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one vertex are defined as vector 
and 
with the specific vertex as the origin, then the volume of the parallelepiped is the same as the scalar triple product of the vectors, or 
. Suppose that 
, 
, 
. We then have the area of the parallelepiped is ![\[|\text{det}(\begin{bmatrix} c_1 & c_2 & c_3 \\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ \end{bmatrix})|.\]](http://latex.artofproblemsolving.com/e/b/d/ebd7498b87d69dc4997c77049a0286cfddf18b77.png)
