Say I have a set of $(n-1)$ linearly independent vectors $\mathbf{v}_i$ of dimension $n$ with entries $\pm1$. I am interested in finding the $n-$dimensional vector $\mathbf{u} $which is normal to the hyperplane spanned by the $\mathbf{v}_i$. In other words, $\mathbf{u}$ is orthogonal to each of the $\mathbf{v}_i$, where $\mathbf{u}$ is unique up to multiplicative constant.

I have two questions:

1) Is it true that $\mathbf{u}$ is also a vector with entries $(0, \pm 1)$ with an arbitrary pre-factor (e.g. $\mathbf{u}=\alpha[-1,1,0]$)? Can this be proven?

2) I know that $\mathbf{u}$ can be calculated with standard linear algebraic techniques (e.g. Gaussian elimination), but if property 1) is true, is there a computationally faster trick to getting the answer?

Thank you for your help!