# How to introduce perpendicular or congruence of angles in affine space

$n$-dimensional affine point-vector space is a pair $\mathbb A^n = \langle \mathbb A, V^n \rangle$, where $\mathbb A$ is an arbitrary set, which elements are called points of affine space, $V^n$ is an $n$-dimensional real vector space together with function $\mathbb A \times \mathbb A \to V^n$, denoted $(A,B)\mapsto \vec{AB}$, such that the following axioms hold:

1. $\vec{AB}+\vec{BC} = \vec{AC}$ for any $A,B,C \in \mathbb A$,
2. For any $A \in \mathbb A$ and for any $r \in V^n$ there exists a unique $B \in \mathbb A$ such that $\vec{AB} = r$.

In my book it is written that $\mathbb A^3$ is a simpliest model of our physical space (Golubev, Foundations of theoretical mechanics). Then author uses such notions as "two lines are perpendicular". My question is how to introduce a perpendicular in affine space? Hilbert in his Foundations of Geometry introduce a notion of right angle as the angle that is congruent to its complement. Then two lines are perpendicular if their intersection provide us four right angles. What is the way to introduce relation of congruence of angles in affine space then?

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You cannot define angles in affine space, in the sense that this is impossible to do in a way invariant under the affine group. –  Mariano Suárez-Alvarez May 20 '13 at 8:16
Hilbert is able to do what he does because his constraints are simpler: his notion of angle need only be invariant under the euclidian group. –  Mariano Suárez-Alvarez May 20 '13 at 8:18
@MarianoSuárez-Alvarez hm? Why I can't, for example, introduce a scalar product in $V^n$ and then measure angle between lines $AB$ and $AC$ as $\mathop{\mathrm{arccos}} \frac{\langle \vec{AB}, \vec{AC} \rangle}{|\vec{AB}| |\vec{AC}|}$? –  Nimza May 20 '13 at 8:20
No scalar product is invariant under the affine group. –  Mariano Suárez-Alvarez May 20 '13 at 8:24
@MarianoSuárez-Alvarez There is nothing said about the action of affine group in my book. If I introduce such notion of angle, what will happen? Axioms I've written in my post will be inconsistent or what? As I remember school level geometry, all axioms from my post worked well for the space we considered –  Nimza May 20 '13 at 8:28
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