Hello, You should clarify your wording. What do you mean by "if their sum is not zero" ? Do you mean that we then keep the two initial vectors or we just eliminate them without replacing them by their sum (which is just one point) ? Also can you please define what you call two "different" methods ? Thanks, -- Julien Santini

We have some allowed operations in each operation we can add two vectors that their sum is not zero vector (with problem conditions); And we can move a vector on a line that it's on it. You see we can do these allowed operations in many methods to reach to a single vector. You must prove that all these vectors are on the same line I wonder why you don't understand what I said. I think everything is clear!!

Omid Hatami wrote: I wonder why you don't understand what I said. I think everything is clear!! The grammar, for starters.

I think this problem reduce to showing the operation become associative (eg. $n=3$), and this we can represent as a triangle geometry problem. After associativity then rest is simple

The sum of the vectors is invariant. Right??? Or I'm wrong?