In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined. Properties The scalar triple product is unchanged under a circular shift of its three operands (a, b, c): Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product: Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product: The scalar triple product can also be understood as the determinant of the matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose): If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume. If any two vectors in the scalar triple product are equal, then its value is zero: Also: The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a Gram determinant. The ratio of the triple product and the product of the three vector norms is known as a polar sine:which ranges between -1 and 1. Scalar or pseudoscalar Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the orientation of the frame or the