I have the following definitions:
Given a vector space $V$ over a field $k$, we can define the projective space $\mathbb P V = (V \backslash \{0\}) / \sim $ where $\sim$ identifies all points that lie on the same line through the origin.
A projective subspace $\mathbb P W$ of $\mathbb P V$ is of the form $\pi(W \backslash \{0\})$, where $\pi$ is the residue class map and $W$ is a vector subspace of $V$. Define $\mathrm{dim} (\mathbb P V) = \mathrm{dim}( V )- 1$. A line in $\mathbb P V$ is a $1$-dimensional projective subspace.
Now I'm finding it difficult to visualise what a line in projective space actually is. I can understand why any two lines in a projective plane intersect. Suppose I'm in $\mathbb P^3$ and want to write 'an equation' for the line that goes through the points $ p = (1:0:0:0)$ and $q = (a:b:c:d)$. How could I do that? Does my question even make sense? I'm concerned because $\mathbb P V$ isn't actually a vector space, so can I think of points inside it as vectors?
Thanks