Does this hold anywhere close to remotely over finite fields and rings?

Question

We know in two dimensions if $\langle u,v\rangle=1$ where $u,v\in\mathbb R^2:\|u\|=\|v\|=1$ holds and $\langle u,a\rangle=0$ holds at some $a\in\mathbb R^2$ then $\langle v,a\rangle=0$ holds.

What if instead we have $\mathbb K^2$ where $\mathbb K$ is a ring with zero divisors or $\mathbb K=\mathbb F_{p^t}$?

Answer 1

Over $\mathbb{R}$ this is false, with for example $u=(1,0)$, $a=(0,1)$ and $v=(1,1)$.