📂Geometry

Parallelepiped

Definition

Simple Definition

A hexahedron in which all faces are parallelograms is called a parallelepiped.

Linear Algebraic Definition

For three distinct vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ in a 3-dimensional coordinate space, the following set is called a parallelepiped.

$$ R = \left\{ \lambda_{1}\mathbf{a} + \lambda_{2} \mathbf{b} + \lambda_{3} \mathbf{c} : 0\leq \lambda_{1},\lambda_{2},\lambda_{3}\leq 1 \right\} $$

Explanation

It is a 3-dimensional extension of a parallelogram. It is defined in the same way even when extended to $n$ dimensions.

Properties

Volume

The volume formula is as follows.

$$ \begin{align*} V &= A \times h = \begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} \\[2em] &= abc \sqrt{1 + 2\cos\alpha \cos\beta \cos\gamma - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma} \end{align*} $$

Here, $A$ is the area of the base, $h$ is the height, $a$, $b$, $c$ are the magnitudes of the vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, respectively, and $\alpha = \angle(\mathbf{b}, \mathbf{c})$, $\beta = \angle(\mathbf{a}, \mathbf{c})$, $\gamma = \angle(\mathbf{a}, \mathbf{b})$ are the angles between the vectors.