Angle between two given vector is small. Can we permute coordinates of them such that new vectors be orthogonal? [closed]

Question

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 $\sigma$?

Answer 1

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 $x_1 y_2+x_2 y_1 + x_3 y_3=0$.