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?
One interpretation is that one of the vectors can not help in constructing the other vector by linear combinations. The closest fit will always be for length 0 of that vector in any linear combination. So if we geometrically view a vector ${\bf v} = [x,y,z]$ as constructed by summing a set of other vectors ${\bf e}_k$:
$${\bf v} = \sum_{\forall i} s_i {\bf e}_i$$
For any ${\bf e}_k = [a,b,c]$ perpendicular to $\bf v$ there would be no point in having $s_k \neq 0$.