Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two *unit* vectors such that $\sum\_{i}{x\_i}=\sum\_{i}{y\_i}=0$ 


$$ x\_{1}y\_{1}+x\_{2}y\_{2}+\cdots +x\_{n}y\_{n} \gt 1-\frac{1}{n} .$$ 


Can we prove that
$$x\_{1}y\_{\sigma(1)}+x\_{2}y\_{\sigma(2)}+\cdots +x\_{n}y\_{\sigma(n)} \neq 0$$ 


for all permutations $\sigma$? 



Angle between two given vector is small. Can we permute coordinates of them such that new vectors be orthogonal?
No. Let 
$$\vec{x} = \vec{y} = \frac{1}{\sqrt{12+6 \sqrt{3}}} (-2-\sqrt{3},1,1+\sqrt{3}).$$
So $\vec{x} \cdot \vec{y} = 1$, but $x\_1 y\_2+x\_2 y\_1 + x\_3 y\_3=0$.


