Scalar Triple Product

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

Scalar Triple Product

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.

Mixed Product

Examples

Calculate the triple product of the following vectors:

Triple Product Example

Triple Product Calculations

Triple Product Solution


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:

Mixed Product Example

Mixed Product Solution


Volume of a Tetrahedron

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

Volume of a Tetrahedron

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 Example

Triple Product Example

Triple Product Example

Volume of a Tetrahedron

Triple Product Calculations

Triple Product Solution


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.

Triple Product Properties

Triple Product Properties

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