Geometric applications

The area of a parallelogram with adjacent sides $\overrightarrow{u}$ and $\overrightarrow{v}$ is

$A_{\text{llgram}}=|\overrightarrow{u}\times \overrightarrow{v}|\text{.}$

A triangle is really just half of a parallelogram, so the area of a triangle with sides $\overrightarrow{u}$ and $\overrightarrow{v}$ (the third side doesn’t matter) is

$A_{\text{tri}}=\frac{1}{2}|\overrightarrow{u}\times \overrightarrow{v}|\text{.}$

The volume of a parallelepiped (a solid body of which each face is a parallelogram) with edges $\overrightarrow{u}\text{,}$ $\overrightarrow{v}\text{,}$ and $\overrightarrow{w}$ sharing a vertex is

$V_{\text{llpiped}}=|\overrightarrow{u}\times \overrightarrow{v}\cdot \overrightarrow{w}|\text{.}$

The order of the three vectors does not matter. Like always, the cross product must be evaluated before the dot product. The absolute value bars here actually mean absolute value, not magnitude, since the dot product produces a scalar. This value is sometimes negative, but volume must be nonnegative. Notice the similarity to the coplanarity test: if three vectors form a parallelepiped of zero volume, they are coplanar.