For some 3 dimensional vector space over a finite field $F_{p^k}$ how can I determine (and visualise) the elements which are in a plane which is perpendicular to the vector $(1, 1, 1)$? For an $n$ dimensional vector space over $F_{p^k}$ how can I determine/visualise the points which are in the $n-1$ dimensional space perpendicular to the vector $(1,1,\dots,1)$?
In the case of a 2 dimensional vector space the line perpendicular to the unit vector $(1,1)$ which passes through the point $(1,1)$ can be visualised as a diagonal line from $(0, p^k - 1)$ to $(p^k - 1, 0)$ but I'm struggling to visualise even the 3 dimensional case.