I'm a bit stuck with this exercise from a script I'm reading, and I'm not very familiar with projective $n$-space yet. The problem:
Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $p\in\mathbb{P}^3\smallsetminus(L_1\cup L_2)$. Show that there is a unique line $L\subseteq\mathbb{P}^3$ meeting $L_1$, $L_2$, and $p$ (i.e. such that $p\in L$ and $L\cap L_i\neq\varnothing$ for $i=1,2$).
To be honest, I already have a problem with the term 'line'. As I take it, a line in $\mathbb{P}^3$ should be something cut out by two degree-1 polynomials (homogeneous, since it wouldn't be well defined otherwise, right?). But what are disjoint lines in projective space? As far as I understood it, two distinct lines should always intersect in exactly one point there, so how can they be disjoint? Can it be that these two statements mean a different kind of 'line'?
Any explanation of this, hints, or even a solution would be very appreciated. Thanks in advance!