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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: 258


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