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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 (1;2;1), (0;1;5), (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: 1604

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