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:

The triple product of coplanar vectors equals zero.

The triple product equals

, because .

Thus, the coplanarity condition is

Example 2. Show that the four points А(1;2;–1), В(0;1;5), С(–1;2;1), and D(2;1;3) belong to the same plane.

B C A D

Compose the vectors ={–1;–1;6}, ={–2;0;2}, ={1;–1;4}.

To show that they are coplanar, we find the triple product

.

Thus, the four points belong to the same plane.

Date: 2015-01-02; view: 1727