I have a doubt regarding the significance of a force on a wire. Well, first of all, I know that if I have a particle and if there are several forces acting over it, then we can compute one total force $F$ that gives the same effect as the combination of the several forces, and this force is just the vector sum.
Well, this is pretty good: a vector depends on the point it's being applied (since it's an element of the tangent space at the point), so that since all the forces are on the same point (the same tangent space) we can take their sum and get another thing acting on the same point. This is pretty clear and simple to understand.
My doubt is when we have a wire for example. For a wire normally we use the relationship:
$$F=\int_\gamma i \ \gamma'(t)\times B(\gamma(t)) \ dt$$
In other words, we parametrize the wire with some curve $\gamma$, and we integrate $i \gamma' \times B$ over the wire to get the "force on the wire". But now this is confusing me, we are summing vectors at different points, and getting a vector that I don't know where's located. What I mean is: while when working with particles it makes sense to add the forces and use the total force on the same particle, with a wire this is kind of confusing, because it has length, so what should really mean "a force on a wire" since it's not just a point?
Thanks in advance for the help.