the scalar triple product

Three arrows sprouting from a common corner sketch out a slanted box — a parallelepiped. How much space does that box enclose? The scalar triple product answers in a single number, and it even tells you, by its sign, whether the three arrows form a right-handed or left-handed set. It is the dot product and the cross product working as a team.

For three vectors u, v, w, the scalar triple product is u . (v x w): first cross v and w to get a vector perpendicular to both, then dot that with u. The result is a single number (a scalar). Its absolute value is the volume of the parallelepiped with edges u, v, w; its sign records orientation (positive for a right-handed triple, negative for left-handed). A clean way to compute it is the 3-by-3 determinant whose rows are the components of u, v, w. A beautiful symmetry holds: u . (v x w) = v . (w x u) = w . (u x v) — you may cyclically rotate the three vectors without changing the value (swapping any two flips the sign). The product is zero exactly when the three vectors are coplanar, since a flat box has no volume.

This single number does a lot of geometric work: it tests coplanarity (zero means the three vectors lie in one plane, hence four points A, B, C, D are coplanar when AB . (AC x AD) = 0), it gives volumes of parallelepipeds and, divided by six, of tetrahedra. The most common slip is bracketing: u . (v x w) makes sense, but (u . v) x w does not — you cannot cross a number (u . v) with a vector. The notation must keep a cross product feeding into a dot product, never the reverse.