Let $\mathbb R^n$ be an Euclidean space with the natural metric, and let $V$ be a subspace such that \begin{equation} V\cap\{(x_1,\cdots,x_n)\colon x_i>0\;\forall i\}=\emptyset \end{equation} and \begin{equation} V\cap\{(x_1,\cdots,x_n)\colon x_i\geq0\;\forall i\}=\{(0,\cdots,0)\} \end{equation}
Let $W=V^\perp$ and can someone help to prove \begin{equation} W\cap\{(x_1,\cdots,x_n)\colon x_i>0\;\forall i\}\ne\emptyset \end{equation}
Thank you.