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The scalar product of two vectors **r**_{1} and **r**_{2} referred to
the same base system consisting of the three non-coplanar vectors ,, is defined as:

(1) |

In matrix notation it could be written:

(2) |

(3) |

and

If in (3) we assume **r**_{1} = **r**_{2}, we have:

(4) |

(5) |

On the other hand, bearing in mind that , where is the angle between and , we have:

(6) |

(7) |

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.

**Copyright © 1980, 1998 International Union of
Crystallography**