We have the equipollence relation between ordered pairs of points in the physical space as follows: $$\ (A, B) \sim (C, D) \iff [AD] \text{ and } [BC] \text{ have the same midpoint }$$ A vector is the following thing: $$\ \vec{AB} = \{ (X,Y) \text{ } | \text{ }(X,Y) \sim (A,B) \}$$ So, if a vector is a set, how it is possible to say $\ \vec{x} + \vec{y}$? Can you add two sets? And if so, can you explain it to me? And furthermore, does the triangle rule of adding two vectors make sense? What is the proof for this rule?
For who does not know what is the triangle rule of adding two vectors:$$\ \vec{AB} + \vec{BC} = \vec{AC} \text{ } \forall \text{ } A,B,C \text{ points in the physical space } $$
The triangle rule can be taken as a definition of vector addition for "physical" vectors.
If we take a pair of directed line segments $(A,B)$ and $(B,C)$ (note that they must have point $B$ in common) then we can define the sum of these line segments as
$(A,B) + (B,C) = (A,C)$
To extend this definition to vectors, note that although you cannot directly add two sets $\vec{AB}$ and $\vec{BC}$, you can add representatives from these sets. But we have to show that addition is well defined if we replace $(A,B)$ by $\vec{AB}$ and $(C,D)$ by $\vec{CD}$. In other words, if we take any $(X,Y) \in \vec{AB}$ and $(Y,Z) \in \vec{BC}$ then
$(X,Y) + (Y,Z) = (X,Z) \in \vec{AC}$
regardless of our choice of $X$, $Y$ and $Z$.