Vector axiomatics

vector point axiomatics

The axiomatics of an $ n $-dimensional affine space $ R ^ {n} $, the basic concepts of which are "point" and "vector" ; the connection between them is realized by establishing a correspondence between a pair of points and a uniquely defined vector. The following axioms are valid.

I) The set of all vectors of $ R ^ {n} $ is an $ n $-dimensional vector space $ V ^ {n} $.

II) Any two points $ A $ and $ B $, given in a definite order, define a unique vector $ \mathbf u $.

III) If a vector $ \mathbf u $ and a point $ A $ are arbitrary given, there exists only one point $ B $ such that $ \mathbf u = \vec{AB} $.

IV) If $ \mathbf u _ {1} = \vec{AB} $ and $ \mathbf u _ {2} = \vec{BC} $, then $ \mathbf u _ {1} + \mathbf u _ {2} = \vec{AC} $.

The pair "point A and vector u" is called "the vector u applied at the point A" (or "fixed at that point" ); the point $ A $ itself is said to be the origin of the vector $ \mathbf u $ applied at it, while the point $ B $ which is uniquely defined by the pair $ A, \mathbf u $ is said to be the end of the vector $ \mathbf u $ (applied at $ A $).

An arbitrarily given vector $ \mathbf u $ generates a completely defined one-to-one mapping of the set of all points of $ R ^ {n} $ onto itself. This mapping, which is known as the translation of $ R ^ {n} $ over the vector $ \mathbf u $, relates each point $ A \in R ^ {n} $ to the end $ B $ of the vector $ \mathbf u = \vec{AB} $.

References

[1]	P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian)
[2]	P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)

Comments

Cf. also (the editorial comments to) Vector or [a1].

References

[a1]	M. Berger, "Geometry" , I , Springer (1987) pp. Chapt. 2