# Scalar Triple Product

The scalar triple product or mixed product of the vectors , and is denoted by [, , ] and equals the dot product of the first vector by the cross product of the other two.

The mixed product of three vectors is equivalent to the development of a determinant whose rows are the coordinates of these vectors with respect to an orthonormal basis.

#### Examples

Calculate the triple product of the following vectors:

### Volume of a Parallelepiped

Geometrically, the absolute value of the triple product represents the volume of the parallelepiped whose edges are three vectors that meet in the same vertex.

#### Example

Find the volume of the parallelepiped formed by the vectors:

### Volume of a Tetrahedron

The volume of a tetrahedron is equal to 1/6 of the absolute value of the triple product.

Calculate the volume of the tetrahedron whose vertices are the points A = (3, 2, 1), B = (1, 2, 4), C = (4, 0, 3) and D = (1, 1, 7).

### Triple Product Properties

1. The triple product does not change if the order of its factors are circularly rotated, but changes sign if they are transposed.

2. If three vectors are linearly dependent, the triple product is 0.