## Volume of a Parallelepiped

For a parallelepiped formed by three vectors
$\vec{a}, \: \vec{b}, \: \vec{c}$
the area of the base - formed by vectors
$\vec{a}, \: \vec{b}$
is the magnitude of the vector or cross product of
$\vec{a}$
and
$\vec{b}$
.
$Area \: of \: Base = \| \vec{a} \times \vec{b} \|$

$\vec{a} \times \vec{b}$
is a vector perpendicular to both
$\vec{a}$
and
$\vec{b}$
).
If the third vector defining the parallelepiped is
$\vec{c}$
then the volume of the parallelepiped is given by the the area of the base times the projection of
$\vec{c}$
onto the vector
$\vec{a} \times \vec{b}$

$Volume=\| \vec{a} \times \vec{b} \| \vec{c} \cdot ( \frac{\vec{a} \times \vec{b}}{\| \vec{a} \times \vec{b} \|} )=(\vec{a} \times \vec{b}) \cdot \vec{c}=\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|$