What is the difference between $0$ and $\vec{0}$? If we take the approach underlined by Peano axioms, we get:
$$a + (-a) = 0.$$
Likewise, we can get the same thing in Euclidean plane:
$$|AB| + (-|AB|) = 0,$$
where $A$ and $B$ are points, and $|AB|$ is the length of the line segment $AB$ (thus the "operation" $AB - AB$ has no meaning, as we have not ascribed any attributes to the line segment).
On the other hand,
$$\vec{AB} - \vec{AB} = \vec{AB} + \vec{BA} = \vec{0}.$$
Well, what is the difference between $0$ and $\vec 0$ ? I think that a concrete definition of both should clear my confusion up. I understand that $\vec 0$ cannot be used in the same context as $0$ (because the former arises from the definition of vector), but I still don't have in-depth understanding of the concept of $\vec 0$. In Euclidean plane both $0$ and $\vec 0$ are points, right? So what's the difference then?

$\vec{0}$ is a vector while $0$ is a number
