If I have a car,
Alan can apply a (horizontal) force on the back bumper on the right
side.
Bob can push on the trunk in the middle.
Carl could open the driver's door and push on the frame.
Dan could grab the passenger door handle.
I suspect that students will intuitively think of these as "different
forces" even if the people are pushing "as hard" as the others. F(A),
F(B), F(C) or F(D) could each be applied independently of the other.
Each has the same effect, but it clearly matters to Alan if he is
actually pushing or not. Heck, we even gave them different names --
"clearly" they are different forces.
But the effect of all of these "different forces" is the same. The car
will accelerate the same in each case (ignoring minor details of the
wheels, suspension, etc). The power of vector notation is that we have
a formalism for saying all these forces are identical. We can draw the
vector anywhere we want and it produces the same effect. Thus all of
these "different vectors" are indeed the same at some level because they
produce the same result. Once we establish that all of these are "the
same" we can draw the vectors to suit our convenience -- for example,
tail to tail to emphasize what object they are acting on, or tip to tail
to emphasize the net force.