Scalar Triple Product (Parallelepiped Volume)

Cross two vectors to get a third whose length is the base area and whose direction is the base's normal, then dot that with the remaining vector to pick out the height — the product |a·(b × c)| is the volume of the box the three vectors lean out into. It is also, term for term, the determinant of the 3×3 matrix holding those vectors as rows, which is the cleanest way to see why a zero determinant means the rows are coplanar: the box is flat and holds nothing.

The three edge vectors (2, 0, 0), (0, 3, 0) and (0, 0, 4) give a rectangular box of 24 m³, which the formula returns as 24 000 L. Less obviously, (1, 2, 3), (0, 1, 4) and (5, 6, 0) span a badly skewed parallelepiped of exactly 1 m³. Lagrange used triple products in his 1773 work on the attraction of ellipsoids, and the operation is still the standard test in mesh software for whether a tetrahedron has been built inside-out — one sixth of the triple product is the tetrahedron's volume, and a negative sign means its vertices were listed in the wrong order.