trippelprodukten

Trippelprodukten, or the triple product, is a term used to describe an operation that combines three operands to produce a result. In mathematics and related fields, the two most common instances are the scalar triple product and the vector triple product. These constructions appear in linear algebra, geometry, and physics, and they connect to determinants, volumes, and orientations.

The scalar triple product is defined for vectors a, b, and c in three-dimensional space as [a, b, c] = a · (b × c). This value equals the determinant of the matrix with a, b, c as its columns. The scalar triple product is the signed volume of the parallelepiped spanned by the three vectors. It is invariant under cyclic permutations [a, b, c] = [b, c, a] = [c, a, b], and it changes sign when any two vectors are swapped.

The vector triple product is given by a × (b × c) and satisfies the bac-cab rule: a × (b × c) = b(a · c) − c(a · b). This identity expresses a cross product of a cross product as a linear combination of the two original vectors and is widely used in mechanics, computer graphics, and vector calculus.

Beyond these, the triple product is an instance of a multilinear form in three arguments and generalizes to higher-order tensors. It provides insight into volumes, orientations, and the interplay of three directions in space. In various languages, trippelprodukten is used as the term corresponding to the English “triple product.”