Consider the hypersurface $X$ in $\mathbb{P}^2(\mathbb{C})\times \mathbb{P}^2(\mathbb{C})$ defined as the zero locus of $$ X:Z(f)= (y_1y_2+y_0^2)x_0+y_1^2x_1+y_2^2x_2=0$$ with $(x_0,x_1,x_2;y_0,y_1,y_2)$ homogeneous coordinates. The variety is smooth at the point $p=(0,1,0;0,0,1)$.
Consider $\phi: Y\to X$ the blow-up of $X$ along $p$. The preimage $\phi^{-1}(p)$ is the exceptional divisor $E$, which in this case is isomorphic to $\mathbb{P}^2$ (I've computed that the tangent space $T_{X,p}$ is isomorphic to $\mathbb{C}^3$, and is given by the $x_2=0$).
Question: I know that the blow-up separates the tangent directions to the point we are blowing-up. But how do I find explicitly a correspondence between $T_{X,p}$ and $E$? Obviously $E\sim (T_{X,p}\setminus \{0\})/\sim$, but I would like an explicit example. For instance, consider the line given by the zero locus $l: Z(f, x_0,y_2)$. Which point in $E$ correspond to $l$?
Consider the affine chart $U$ of $\mathbb P^2\times\mathbb P^2$ given by $x_1=y_2=1$ with coordinates $(x_0,x_2,y_0,y_1)$. The intersection $X\cap U$ of $X$ with this chart is given by equation $(y_1+y_0^2)x_0+y_1^2+x_2=0$, hence it is isomorphic to $\mathbb A^3$ with coordinates $(x_0,y_0,y_1)$. And $\phi^{-1}(X\cap U)$ is isomorphic to the blow-up of $\mathbb A^3$ at the origin. The line $l$ in $\mathbb A^3$ is given by $x_0=y_1=0$, so, if we consider the blow-up of $\mathbb A^3$ at the origin as a subvariety of $\mathbb A^3\times\mathbb P^2$, where $\mathbb P^2$ has coordinates $(x_0':y_0':y_1')$, then the point of $E$ corresponding to $l$ is $((0,0,0),(0:1:0))$.