I seem to be confused about the nature of forces as vectors, in the basic Newtonian mechanics framework.
I know what a vector is as a mathematical object, an element of $R^3$. I understand that if a vector is drawn in a usual physical way as an arrow in space, it can be seen as a mathematical vector by translating it to begin at 0 and seeing where the arrow tip ends up. Generally it seems the word "vector" is used in such a way that a vector remains the same vector if it's translated arbitrarily in space (always corresponding to the same mathematical vector).
But now let's say I have a solid object, maybe a metal cube, with some forces acting on it: I push at it with a stick in the center of one facet, it's held by a rope in a different corner, etc. To specify each force that is acting on the cube it doesn't seem enough to specify the vector: I also need to specify the place of application. The cube behaves differently if I push it in the center as opposed in the corner etc.
I'm reading through J.P.Den Hartog's Mechanics that teaches me how to find the resultant force on the cube. I need to sum forces one by one using the parallelogram law, but I should always be careful to slide each force along its line of application, until two forces meet. I could just translate them all to start at the same point and add, but then I won't find the right resultant force, only its direction and magnitude; I will still need to find its line of application (maybe using moments etc.)
So let's say I'm calculating the resultant force "the right way": by sliding arrows along their lines until tails meet, adding, repeating. What am I doing mathematically? (it's not vector addition, that would correspond to just translating them all to 0 and adding) What mathematical objects am I working with? They seem to be specified with 4 free parameters: 3 for direction/magnitude of the vector and 1 more to displace it to the correct line of application; the location at the line of application seems irrelevant according to the laws of statics.