Triple Scalar Product

The triple scalar product of the vectors a, b, and c is the result of the scalar multiplication of the first of these vectors by the vector product of the second and third vectors: a·(b × c). It is often written [abc] or (abc).

The numerical value of the triple scalar product is equal to the volume of the parallelepiped of which the vectors a, b, and c are coterminal sides; the volume is taken with a plus sign if a, b, and c are oriented in the same way as the unit coordinate vectors i, j, and k, and with a minus sign otherwise. If a, b, and c are written in the forms a = a₁ i + a₂ j + a₃k, b = b₁i + b₂j + b₃ k, and c = c₁ i + c₂ j + c₃ k, then the triple scalar product is the determinant of the coefficients of i, j, and k. A triple scalar product remains unchanged under a cyclic permutation of a, b, and c; in the case of a noncyclic permutation, the product changes in sign.