According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are:
G1: Every line contains at least 3 points
G2: Every two points, A and B, lie on a unique line, AB.
G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).
Now let $V$ be a vector space over a field $K$. We denote by $P(V)$ the set of one-dimensional subspaces of $V$. If $V$ is finite dimensional, this is the usual definition of a projective space over $K$. We say a two-dimensional subspace of $V$ a line of $P(V)$. Then it is natural to expect that points and lines of $P(V)$ satisfy the above axioms. I think it is easy to prove G1 and G2. But how can we prove G3?
Claim: $AB$ intersects $CD$ if and only if the underlying vectors of $A,B,C,D$ are linearly dependent.
(in fact, if $AB \neq CD$, they span a space of dimension $3$)