Lines in Fields of Three

Suppose the elements of $\mathbb{F}_3$ are of the form $-1$ , $0$ and $1$ , so that elements of $\mathbb{F}_3^2$ are pairs of the three elements defined previously. One can represent this space geometrically as a $3 \times 3$ grid, with the origin being the middle square in this grid. It should be clear then that there should be four lines going through this origin, with direction vectors $\left(1,0\right)$ , $\left(1,1\right)$ , $\left(-1,1\right)$ and $\left(0,1\right)$ .

P.S. An alternative representation of this problem is to consider how many possible winning combinations there are in Tic-Tac-Toe that involves the middle square.