I have just started to study vector bundles, and I just know the definition, i.e. a vector bundle over a variety $X$ is a variety $F$ with a map $\pi:F\rightarrow X$ and a covering by open sets, that for each opens set accepts a map $\psi_i:\pi^{-1}(U_i)\rightarrow U_i\times K^r$, and the composition of $\pi$ with this map is the projection. And also there is transition functions that are linear. The definition is pretty similar with the tangent bundle, so it seems fine.
But the text suggests an exercise after the definition, and I am having trouble with it. It asks to construct a rank two vector bundle over the smooth quadric $X\subset\mathbb P^4$ defined by $x_0x_3+x_1x_4+x_2^2$ by gluing the local equations of the line $L$ defined by $x_0=x_1=x_2=0$.
So I know that looking to the local equations I will be able to obtain the matrix of transition, but I am afraid that I cannot find the local equation (is it defined locally by the ideal $(x_0,x_1,x_2)/(x_0x_3+x_1x_4+x_2^2)$?)
Any hint in how to procede is appreciated.
Thanks in advance.