Let us suppose to have a finite set of vectors $S=\{v_1,\ldots,v_m\}$ in $\mathbb{R}^n$ (with $m \gg n$ in general). I need to find a vector $x \in \mathbb{R}^n$ that is NOT perpendicular to any vector in $S$. The existence of such a vector $x$ is guaranteed, moreover almost every vector in $\mathbb{R}^n$ satisfies this property. But I need to find an algorithm to determine a vector with these properties, I cannot close my eyes and choose.
Any ideas?
EDIT1: in my case, the vectors in $S$ have some "symmetries" in the sense that they are generated by permutation and change of signs of a few vectors in $S$
EDIT2: $v_1+\ldots+v_m=0 \in \mathbb{R}^n$
from Hot Weekly Questions - Mathematics Stack Exchange
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