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 À(1;2;–1), Â(0;1;5), Ñ(–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: 20150102; view: 1681
