I should begin by saying that I'm only beginning to study $11$th grade Physics. Recently I figured out that two vectors which gets added up to a null vector, must at most lie in a line. Well that makes perfect sense, so I thought about $3$ vectors and figured that they must at most lie in a plane. Also $4$ vectors when added up to a null vector must at most lie in $3$D space. I then thought about a single vector. Naturally I can't add a vector to nothing and get $0$ vector right? So that means it must be the null vector itself. So that means A single vector must lie in a dimensionless point to give a null vector.
So I hypothesise that n vectors when added up to a null vector must lie in $n-1$ spatial dimensions. That is iff $$v_1+v_2+v_3....+v_n=0$$ Then $v_1 ,v_2,v_3 ,v_4 ,v_5...,v_n$ must at most lie in $(n-1)$th dimension.
Please put some thought into this and help me understand a method to visualize it if this is correct.