Triple Product of Vectors and its Properties

Definition. Thetriple product of three vectors is the inner product of the third vector by the vector product of the first two vectors; it is denoted by

.

Definition. The vector product of the vector product of the first two vectors and the third vector ones is called the double vector product:

.

Since double vector product is used very rarely, it have been little studied.

Property 1. The triple product of three vectors equals the volume of a parallelepiped spanned by these three vectors.

Corollary. It is easy to derive an expression for the volume of a pyramid from the formula:

S

B

А С

V_рyr= S_base Н = . S_рar Н = V_рar= ,

V_рyr= .

The sign is needed to obtain a positive volume.

Property 2. Triple product is commutative, and

.

Property 3. A constant multiplier of any vector can be factored out of scalar triple product:

.

Triple Product in Coordinates. Given three vectors , , and , let us express the triple product of these vectors in terms of their coordinates. Consider the triple product

.

The vector product equals

.

Taking its inner product with , we obtain

;

 

this is a third – order determinant expanded along the last line, i.e.,

.

Thus, the triple product of three vectors equals the third – order determinant of the composed of the coordinates of these vectors.

Example 1. Determine the volume of a pyramid ABCD from the coordinates of its vertices.

 

D (1;5;2) B (–1;1;3) A (1;2;0) C (0;2;–3)

Compose the vectors   , , .

Let us find the volume of a pyramid by the formulas proved above: