I apologize again for a dumb question!
To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined the projective space $PF^3$ to be $F^3 \setminus \{(0,0,0)\}$, where we identify points which lie on the same line through the origin. The author I'm reading from has noted that we may identify $F^2$ within $PF^3$ under the identification $(x,y) \rightarrow [(x,y,1)]$, where the brackets indicate the class of the element $(x,y,1)$ in $PF^3$. So $PF^3$ easily breaks down into two chunks: $F^2 \cup \{ [(x,y,0)]: x, y \in F$ not both $0 \}$. I hear tell of some "line at infinity", and I'm guessing that's the second chunk, which I've been internally visualizing as a semicircle (or a circle with antipodal points identified).
Now $PF^3$ is not a vector space, so I tried to come up with a definition for a line. I basically had three criteria for what the definition ought to be: 1) lines in $F^2$ ought to be preserved under the identification, 2) the line at infinity ought to be a line, 3) I've heard rough descriptions that every line meets the line at infinity.
2) seemed to be the place to start, so I fixed two points $(v_1, v_2), (w_1, w_2) \in F^2$ and considered the line $\mathcal{l}(v,w) := \{ (v_1 + fw_1, v_2 + fw_2): f\in F\}$. This line clearly maps onto the set $\{ [(v_1 + fw_1, v_2 + fw_2, 1)]: f \in F\}$.
At this point, I began to get a little worried, because it didn't seem like this line would ever intersect the line at infinity (unless I'm wrong about what the line at infinity is supposed to be!).
My questions:
1) What is the definition of a line in $PF^3$? I'm guessing sets of the form $\{[(v_1 + fw_1, v_2 + fw_2, v_3 + fw_3)]: f \in F\}$?
2) What is the "line at infinity" (technically, not some intuitive description about lines meeting at the horizon)?
A line is an equivalence class of triples $(a,b,c)$ with the entries field elements not all $0$. The equivalence is defined in the same way you did it for points
The point $(x,y,z)$ is incident on the line $(a,b,c)$ if $ax+by+cz=0$. We are working with representatives, not equivalence classes as we should, but the definition is independent of the choice of representatives.
The line at infinity is (the equivalence class of) $(0,0,1)$.
Note the beautiful built-in duality.
Sloppily, we can think of a line as determined by an equation $ax+by+cz=0$, with $(a,b,c)\ne (0,0,0)$. Any line has a point at infinity on it. For example, for the line $(2,3,1)$, aka $2x+3y+z=0$, the point at infinity is $(-3,2,0)$.