**Notations**: For a scalar $a\in\mathbb{R}$, denote 
$$\mathrm{sgn}(a)=\left\{
 \begin{array}{l l}
 1 & \mbox{if } a>0\\
 0 & \mbox{if } a=0\\
 -1 & \mbox{if } a<0
 \end{array}.\right.$$
For a vector $r\in\mathbb{R}^n$, $\mathrm{sgn}(r)$ is element-wise.

Now we have a set of vectors $r\_1, \dots, r\_p\in\mathbb{R}^n$, $p<n$. The corresponding sign vectors are $v\_1,\dots,v\_p\in\mathbb{R}^n$ satisfying
$$\mathrm{sgn}(r\_i)=v\_i$$
which implies that the elements of $v\_i$ can only be $1$, $-1$ or $0$.

**Question**: Given a nonzero vector $x\in\mathbb{R}^n$ satisfying $x\perp \mathrm{span}\{r\_1,\dots,r\_p\}$, can we prove $x\notin\mathrm{span}\{v\_1,\dots,v\_p\}$? If not, can you given any counterexamples?