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The scalar product of two vectors r1 and r2 referred to the same base system consisting of the three non-coplanar vectors ,, is defined as:
In matrix notation it could be written:
and rt1 a transposed column matrix ; 9 is the 3 3 matrix of relation (2) and is called a metric matrix or metric tensor, because its elements are dependent both on the length of the base vectors and on the angles formed by them.
If in (3) we assume r1 = r2, we have:
On the other hand, bearing in mind that , where is the angle between and , we have:
Equations (5) and (7) are the rules to obtain the vector lengths and the angles between vectors. The space in which the lengths and the angles between vectors are defined, is called metric space. The metric is given by the G matrix.
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