Scalar Triple Product Calculator — a · (b × c) and Parallelepiped Volume

How it works

a · (b × c) = det[[a₁, a₂, a₃], [b₁, b₂, b₃], [c₁, c₂, c₃]] = a₁(b₂c₃ − b₃c₂) + a₂(b₃c₁ − b₁c₃) + a₃(b₁c₂ − b₂c₁) · parallelepiped volume = |a · (b × c)| · tetrahedron volume = |a · (b × c)| / 6

The scalar (box) triple product dots one vector with the cross product of the other two. Geometrically b × c is perpendicular to the base parallelogram with length equal to its area, and dotting with a multiplies by the height, so the absolute value is the volume of the parallelepiped spanned by the three vectors. It equals the 3 × 3 determinant of the vectors, is zero exactly when they are coplanar, and its sign tells whether (a, b, c) is a right-handed system (OpenStax Calculus Vol. 3 §2.4).

Worked example

a = (2, 0, 1), b = (1, 3, −1), c = (0, 1, 4). b × c = (3·4 − (−1)·1, (−1)·0 − 1·4, 1·1 − 3·0) = (13, −4, 1). a · (b × c) = 2·13 + 0·(−4) + 1·1 = 27. The parallelepiped has volume 27, the tetrahedron 27/6 = 4.5, and the positive sign means the vectors form a right-handed system.

Sources: OpenStax Calculus Vol. 3 — 2.4 The Cross Product (triple scalar product) · Wolfram MathWorld — Scalar Triple Product

Frequently asked questions