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)