Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as $$ d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|v_y\|} $$ where $v_x$ and $v_y$ are non-zero vectors corresponding to $x$ and $y$, respectively. This expression is essentially the sine of the angle between the two lines in $\mathbb{R}^{n+1}$.
My question is: If we instead work over the function field over a finite field, such as $\mathbb{F}_q[T]$, where there is no natural notion of angle, how can we define a "distance" between points in projective space? As quadratic forms are well studied over the function field, I believe there might be some notion of distance. Any insight is very much appreciated.
