View source for Vector axiomatics

''vector point axiomatics''

The axiomatics of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963702.png" />-dimensional affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963703.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963704.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963705.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963706.png" />.

II) Any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963708.png" />, given in a definite order, define a unique vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v0963709.png" />.

III) If a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637010.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637011.png" /> are arbitrary given, there exists only one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637013.png" />.

IV) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637016.png" />.

The pair  "point A and vector u"  is called  "the vector u applied at the point A"  (or  "fixed at that point" ); the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637017.png" /> itself is said to be the origin of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637018.png" /> applied at it, while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637019.png" /> which is uniquely defined by the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637020.png" /> is said to be the end of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637021.png" /> (applied at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637022.png" />).

An arbitrarily given vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637023.png" /> generates a completely defined one-to-one mapping of the set of all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637024.png" /> onto itself. This mapping, which is known as the translation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637025.png" /> over the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637026.png" />, relates each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637027.png" /> to the end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637028.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096370/v09637029.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Lectures on analytical geometry" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Alexandroff [P.S. Aleksandrov] (ed.)  et al. (ed.) , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR></table>



====Comments====
Cf. also (the editorial comments to) [[Vector|Vector]] or [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)  pp. Chapt. 2</TD></TR></table>