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''vector point 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.
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The axiomatics of an $  n $-dimensional affine space $  R  ^ {n} $,  
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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" />.
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I) The set of all vectors of $  R  ^ {n} $
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is an $  n $-dimensional vector space $  V  ^ {n} $.
  
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" />.
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II) Any two points $  A $
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and $  B $,  
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given in a definite order, define a unique vector $  \mathbf u $.
  
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" />.
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III) If a vector $  \mathbf u $
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and a point $  A $
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are arbitrary given, there exists only one point $  B $
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such that $  \mathbf u = \vec{AB} $.
  
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" />.
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IV) If $  \mathbf u _ {1} = \vec{AB} $
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and $  \mathbf u _ {2} = \vec{BC} $,  
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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 <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" />).
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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 $
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itself is said to be the origin of the vector $  \mathbf u $
 +
applied at it, while the point $  B $
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which is uniquely defined by the pair $  A, \mathbf u $
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is said to be the end of the vector $  \mathbf u $ (applied at $  A $).
  
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" />.
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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 $
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of the vector $  \mathbf u = \vec{AB} $.
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 06:40, 9 May 2022


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
How to Cite This Entry:
Vector axiomatics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_axiomatics&oldid=17121
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article