The STP provides direct volume computation for 3D shapes.

Parallelepiped: Volume = |[a b c]| where a, b, c are the three co-terminal edge vectors. This is the most fundamental volume formula in vector algebra.

Tetrahedron with co-terminal edges a, b, c from one vertex: Volume = $\frac{1}{6}$|[a b c]|. The factor 1/6 arises because a tetrahedron is 1/6 of the parallelepiped.

Tetrahedron with vertices A, B, C, D: Volume = $\frac{1}{6}$|[AB AC AD]|. Choose any vertex as the base and compute the three edge vectors from it.

Regular tetrahedron with edge a: Volume = $a^3$*sqrt$\frac{2}{12}$. This is derived by setting up coordinates and computing the STP.

Triangular prism with base triangle having sides a, b and height h along direction c: Volume = $\frac{1}{2}$|a x b| * |c.n|/|n| where n is the unit normal to the base. If the lateral edges are perpendicular to the base, this simplifies to $\frac{1}{2}$|a x b| * h.

Sign convention: The sign of [a b c] indicates the orientation (right-handed vs left-handed system). For volume, always take the absolute value.

Zero volume: [a b c]=0 means the three edges are coplanar — the solid degenerates to a flat figure. This is the coplanarity test.

JEE tip: The tetrahedron volume formula $\frac{1}{6}$|determinant| is one of the most frequently tested formulas. Always remember the 1/6 factor.