Vector axiomatics

vector point axiomatics

The axiomatics of an -dimensional affine space , 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 is an -dimensional vector space .

II) Any two points and , given in a definite order, define a unique vector .

III) If a vector and a point are arbitrary given, there exists only one point such that .

IV) If and , then .

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

An arbitrarily given vector generates a completely defined one-to-one mapping of the set of all points of onto itself. This mapping, which is known as the translation of over the vector , relates each point to the end of the vector .

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