I am studying 0/1 polytopes from Ziegler's lectures on polytopes https://arxiv.org/pdf/math/9909177.pdf. I found a small part of a proof of a corollary, which I do not understand. Here it is (Corollary 26):
Let $\{0,v_1,..., v_{d-1}\}\subseteq \{0,1\}^d$ be points that span a hyperplane H in $\mathbb{R}^d$, and let $V = (v_1,...,v_{d-1})^t \in {0,1}^{(d-1)\times d}$. Then an equation that defines H is given $c^t x=0$, with $c_i =+-det (V_i)$, where $V_i \in \{0,1\}^{(d-1)\times (d-1)}$ is obtained from V by deleting the i-th column.
I do not understand how this works out. Why does the approach described result in the correct vector c which gives the hyperplane, i.e., why do the determinants give the components of the hyperplane vector c?
Any help is very appreciated. Thanks.