Parallelepiped

Definition

Let $p$ be a point of $\mathbb{R}^3$ , and let $a$ , $b$ and $c$ be linearly independent vectors in $\mathbb{R}^3$ . The parallelepiped with the vertex $p$ and the edges $a$ , $b$ and $c$ is the set $\{\, p + s a + t b + u c \;:\; s, t, u \in [0, 1] \,\}.$

Remarks

The parallelepiped has 8 vertices $p + s a + t b + u c$ with $s, t, u \in \{0, 1\}$ , 12 edges and 6 faces. Opposite faces are congruent parallelograms in parallel planes. Its volume is the absolute value of the scalar triple product $a \cdot (b \times c)$ . In Mathlib, the parallelepiped of a finite family of vectors $v_i$ is the set of the sums $\sum_i t_i v_i$ with each $t_i \in [0, 1]$ . It does not ask for linear independence. A cube and a rectangular cuboid are special parallelepipeds.

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Euclidean vector concept
Parallelepiped definition

Tag	`0K9GBTZ1FX`
URI	`m2://0K9GBTZ1FX`
Kind	definition
Writing	W1 stated
MSC	51M20
Defines	Parallelepiped
Uses	Euclidean vector

Parallelepiped

Definition

Remarks

Reading path

Further reading

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