Vectors added up to give null vector.

Question

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.

Answer 1

You may have heard that two points lie on a line (a 1D space) and three points lie on a plane (a 2D space). If you visualise these with more points, it's possible to see that $n$ points lie on an $n-1$ dimensional space. So consider the origin, and the endpoints of $n-1$ of the vectors. Since we now have $n$ points, these points define an $n-1$ dimensional space containing all the points, including the origin. It's now clear that summing all the vectors must give a point within this space since all vectors lie within it. Since the last vector is the negative of the first $n-1$ vectors added up, it must also lie in this $n-1$ dimensional space.

Answer 2

As we have $\sum_1^n v_i = 0$, we get $v_n = -\sum_1^{n-1} v_i$. Thus, out of the $n$ vectors, at most $n-1$ can be independent of each other (i.e. can take values independent of the other vectors). Hence, they can occupy a space of at most $n-1$ dimensions.