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Volume of a parallelepiped

Three vectors from the same corner span a parallelepiped. The volume is the absolute value of the dot product of the third vector with the cross product of the first two. If the vectors lie along the axes, the volume is simply the product of the lengths. The cross product gives a normal, and the dot product measures how far the third vector sticks out along that normal.

Symbols

Example

$[2, 0, 0]$, $[0, 3, 0]$ and $[0, 0, 4]$ give volume $2\cdot 3\cdot 4 = 24$. The cross product of the first two is $[0, 0, 6]$, and $6\cdot 4 = 24$.

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Part of Mathematics R2: Lines and planes in space.