Suppose there are $M$ known real vectors $$ x^{i}=[x_{1}^{i},x_{2}^{i},\cdots,x_{N}^{i}],i=1,2\cdots,M $$ in N-dimensional vector space and $M\geq 2N$. I want to know whether they span two orthogonal hyperplanes, that is some of these vectors span one hyperplane while the rest span another and these two hyperplanes are mutually orthogonal (I mean their normal vectors are orthogonal, maybe 'vertical' is a better word). So I write the unknown normal vectors to the two hyperplanes as $$ n=[n_{1},n_{2},\cdots,n_{N}]\\ m=[m_{1},m_{2},\cdots,m_{N}] $$ and they are orthogonal $$ m\cdot n=0 $$ Now if $x$ are all on these two hyperplanes then the following equations must have a solution $$ (m\cdot x^{i})(n\cdot x^{i})=0 $$ for all $x^{i}$. Expanding the above equations, $$ (n_{1}x_{1}^{i}+n_{2}x_{2}^{i}+\cdots +n_{N}x_{N}^{i})(m_{1}x_{1}^{i}+m_{2}x_{2}^{i}+\cdots +m_{N}x_{N}^{i})=0\\ \sum_{i=1}^{N}n_{i}m_{i}=0 $$ are multivariate quadratic equations of $n_{i}$ and $m_{i}$. Now my question is: How do I know whether these equations are solvable or not given certain $x^{i}$s? Are there any constraints that can be placed on $x^{i}$s so that these equations have at least one real solution?
Solving multivariate quadratic equations is a NP-hard problem in general. But I think this special form maybe not that hard.
English is not my mother language so I've tried to explain my question as clear as possible. If there are any confusing points in my description, please leave a comment and I'll answer you as soon as I can. Thank you!
Supplement: Maybe an example will help. Consider one vector $x^{1}=[1,1]$ in 2-dimensional real vector space. The above equations should be $$ (n_{1}+n_{2})(m_{1}+m_{2})=0\\ n_{1}\cdot m_{1}+n_{2}\cdot m_{2}=0 $$ Some simple calculation gives $$ \frac{n_{1}}{n_{2}}=-\frac{m_{1}}{m_{2}}=-\frac{m_{2}}{m_{1}}=\pm 1 $$ Because only the orientations of $n$ and $m$ are cared about, the ratios between entries are enough to specify the two vectors. Now if we add another vector $x^{2}=[0,1]$ to the original one-vector set, the equations should be $$ (n_{1}+n_{2})(m_{1}+m_{2})=0\\ n_{2}\cdot m_{2}=0\\ n_{1}\cdot m_{1}+n_{2}\cdot m_{2}=0 $$ This time there is no real solution. It is harder to tell whether there is a real solution for these equations in higher dimensional space.
