Scalar Triple Product: Volume and Coplanarity Guide

The scalar triple product of three vectors a, b, c is the number a · (b × c), written [a b c]. You first take the cross product of b and c, then dot the result with a — the final answer is a single scalar, not a vector.

Its value equals the volume of the parallelepiped, the slanted 3D box whose edges are a, b and c. Because of this, it is the standard tool for computing volumes in 3D coordinate geometry and for testing whether three vectors lie in one plane.

Key Concepts

• Notation: [a b c] = a · (b × c) = (a × b) · c. The dot and the cross can be swapped without changing the value.
• Determinant form: [a b c] equals the 3×3 determinant with rows a1, a2, a3 then b1, b2, b3 then c1, c2, c3. This is the practical way to compute it.
• Cyclic symmetry: [a b c] = [b c a] = [c a b]. Rotating the order in a cycle keeps the value; any other swap flips the sign.
• Volume: |[a b c]| is the volume of the parallelepiped formed by a, b, c, measured in cubic units of the coordinate system.
• Volume of a tetrahedron with coterminous edges a, b, c: V = (1/6) |[a b c]|.
• Coplanarity test: [a b c] = 0 exactly when a, b, c are coplanar — the box collapses flat, so its volume is zero.

Formulas

• Definition: [a b c] = a · (b × c).
• Determinant: the 3×3 determinant of the three component rows.
• Volume of parallelepiped: V = |[a b c]|.
• Volume of tetrahedron: V = (1/6) |[a b c]|.
• Cyclic: [a b c] = [b c a] = [c a b]; anti-cyclic: [a c b] = −[a b c].
• If any two of the vectors are equal or parallel, the triple product is zero.

Worked Example

Let a = (2, 0, 1), b = (1, 1, 0), c = (0, 2, 3). Find the volume of the parallelepiped they form.

1. Compute b × c: x-component = 1×3 − 0×2 = 3; y-component = 0×0 − 1×3 = −3; z-component = 1×2 − 1×0 = 2. So b × c = (3, −3, 2).
2. Dot with a: a · (b × c) = 2×3 + 0×(−3) + 1×2 = 6 + 0 + 2 = 8.
3. Volume = |8| = 8 cubic units.
4. Since the value is not zero, the three vectors are not coplanar.
5. Cross-check with the determinant of rows (2,0,1), (1,1,0), (0,2,3): 2×(1×3 − 0×2) − 0 + 1×(1×2 − 1×0) = 6 + 2 = 8. Matches.

Common Misconceptions

• It is a vector. The name says scalar: a · (b × c) is a number. The vector triple product is the one that returns a vector.
• Any order gives the same value. Only cyclic rotations keep the value; swapping two vectors flips the sign, for example [a c b] = −[a b c].
• Zero means one of the vectors is zero. Zero only means the three are coplanar. Three non-zero vectors lying in one plane give zero.

Real-World Applications

• Computing volumes of parallelepipeds and tetrahedra in 3D geometry problems.
• Testing coplanarity: four points are coplanar when the triple product of three edge vectors from one of them is zero.
• Crystallography: unit-cell volumes are scalar triple products of lattice vectors.
• Physics: flux and phase-space volume calculations use triple products.

The scalar triple product [a b c] = a · (b × c) packs the volume of a 3D box into one number, computed as a determinant. Remember its cyclic symmetry and the zero-means-coplanar test, and you own one of the most-used tools in 3D geometry.