I am given 3 linearly independent vectors $x,y,z \in \mathbb{R}^n$ and I would like to understand which quadrants the orthogonal complement of their span intersects (depending on the coefficients of $x,y,z$). What is the best way to think about this? Any pointers are appreciated.
I've seen at the answers of an other question (What is the maximum number of quadrants in $n$ dimensional space that a $k$ dimensional hyperplane can pass through?) that matroids will likely have something to do with this. What are some good introductory writings on matroids?
The comments in the linked question are suggesting not just matroids, but oriented matroids. The standard reference is Björner et al. Oriented Matroids, but the text is not introductory as it assumes familiarity with matroids. I do not know of any introductory text on oriented matroids, however the material does not have a steep learning curve once you get familiar with (ordinary) matroids.
An introductory text on matroids is Gordon and McNulty, Matroids: A geometric Introduction.
The standard references on matroids are Oxley, Matroid Theory, Welsh, Matroid Theory and White, Theory of Matroids.