Is the product of two vectors of null sum, a vector with null sum?

Question

Let us imagine we have two vectors $A=(a_{1},\ldots,a_{n})$ and $B=(b_{1},\ldots,b_{n})$ of dimension $[1\times n]$ that are only comprised of either $-1$ or $+1$. Furthermore there are as many $-1$ entries as there are $+1$ entries so that: $$ \sum_{i=1}^{n} a_{i} = \sum_{i=1}^{n} b_{i} = 0 \text{.}$$ Is there a way to prove that for any of these two vectors we have: $$\sum_{i=1}^{n} a_{i}\cdot b{i} =0 \text{,}$$ where $\cdot$ is the element-wise product of two vectors.

Answer 1

Sadly, it's false. For $n=2$, take $a=(-1,1)$, $b=(1,-1)=-a$.