In the book on Linear Algebra that I am using, the author defines a line in an arbitrary vector space $V$, given direction $ 0 \neq d \in V $ and passing through $ p \in V$ as $ l(p;d)= \lbrace v\in V| \;\exists t \in \mathbb R,v=p+td\rbrace $.
He then proceeds to show that $ l(p;d)=l(q;d) $ iff $ (q-p) $ is a multiple of $d $. and further that any two distinct points determine a unique line. He later defines 2 lines $ l(p;d_1) , l(q;d_2)$ as parallel if $d_1= \alpha d_2$ for some $ \alpha \in \mathbb R$.
He finally proves Euclid's parallel postulate using these tools. My question is how does this setup still allow for a Non-Euclidean Geometry? Does it have to do anything with the fact that the definition of line here uses a "$ t \in \mathbb R$" which inhibits the validity of this definition? If so, how does one define a straight line using an arbitrary field? I hope I have been able to convey my thoughts precisely enough to warrant an answer.