Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{P}^n$ such that $\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$ (for example, $E$ might be $\mathbb{P}^n\times\mathbb{C}^{m+1}$).

Let $\mathcal{Q}$ be the universal quotient bundle on $\mathbb{P}^n\times\mathbb{P}^m$ of rank $m$, coming from the short exact sequence 
$$
\begin{align}
0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0.
\end{align}
$$

Let $l$ be a line in $\mathbb{P}^n$. Then, in Chow groups, $\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$. 

**Question:** In $CH_1(\mathbb{P}^n\times\mathbb{P}^m)$, do we have  $c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$?, 

where $c_m(\_)$ denotes the $m$-th chern class, and $*$ denotes any point in $\mathbb{P}^m$?