I will ask my question using $\mathbb R^3$ as a reference.
Vectors of the form $[x,y,z]$ perpendicular to $[a,b,c]$ are those where $[a,b,c]\cdot [x,y,z]=0$, that is where $ax+by+cz=0$.The $x,y,z$ components of these vectors must satisfy the equation of the above plane.
At first, I was like that makes sense asthe vector $[a,b,c]$ is perpendicular to the plane $ax+by+cz=0$. But, it kind of doesn't because all those other parallel planes $ax+by+cz=d$ where d is non-zero contain the very same vectors (whose components satisfy $ax+by+cz=0$.) So, what is the meaning geometrically of the plane $ax+by+cz=0$ and the fact that the vector [a,b,c] is perpendicular to it? How would you represent the collection of vectors perpendicular to a given vector or explain it?