Let $G=G(1,3)$ be the grassmanian of lines in $\mathbb{C}P^3$. Let $E$ be the bundle on $G$ whose fiber over $\ell$ are homogeneous linear functions on $\ell$. I am interested in $c(E)=1+c_1(E)+c_2(E)$. By considering Schubert cells on $G$ and the interpretation of top chern class as the dual of the zero locus of a section, I very easily can understend $c_2(E)$ and fit it into my calculations.
I know that the answer should be that $c_1(E)$ is just the generator of $H^2(G;\mathbb {Z})$, that is the poincare dual of the closure in $G$ of the set of all lines intersecting a fixed line $L$, but I don't know how to prove it...
Any hints as to how I can determine it would be appreciated. The question also of course easily generalizes to different grassmanians and it would be interesting to see a general answer.
Thank you!