Dreifachprodukts

Dreifachprodukt is a mathematical term used in German to refer to constructions that combine three quantities, most commonly the scalar triple product and the vector triple product of vectors in three-dimensional space. The form Dreifachprodukts is not standard in technical usage; the singular is usually Dreifachprodukt and the plural Dreifachprodukte.

Scalar triple product. For vectors a, b, and c in R3, the scalar triple product is a · (b × c). It equals the determinant of the 3×3 matrix with a, b, c as rows or columns, det[a; b; c]. Its magnitude equals the volume of the parallelepiped spanned by a, b, c, and its sign encodes orientation. The scalar triple product is invariant under cyclic permutations: a · (b × c) = b · (c × a) = c · (a × b). It vanishes when the three vectors are linearly dependent.

Vector triple product. The vector triple product is a × (b × c), which can be rewritten using the BAC-CAB rule as a × (b × c) = (a · c) b − (a · b) c. This identity is a standard tool in vector calculus and physics, where cross products often appear in torque and angular-momentum expressions.

Notation and usage. In texts, the term is typically distinguished as skalarer Dreifachprodukt (scalar triple product) and Vektor-Dreifachprodukt (vector triple product). Beyond three dimensions, a true cross product does not generalize universally; the scalar triple product generalizes as a determinant, and the vector version is tied to the three-dimensional cross product.