In Vakil's November 2017 set of notes, section 18.4 deals the universal plane conic, defined as follows. Fix a base field $k$, and everything takes place over it. Consider $\mathbb{P}^2$ with homogeneous coordinates $x_i$, $0\leq i\leq 2$, and $\mathbb{P}^5$ with homogeneous coordinates $a_{ij}$, $0\leq i\leq j\leq 2$. Then let $\mathcal{C}$ denote the closed subscheme of $\mathbb{P}^2 \times \mathbb{P}^5$ defined by the equation $$\sum_{0\leq i\leq j\leq 2} a_{ij} x_i x_j = 0.$$ This comes equipped with a natural projection map $\pi : \mathcal{C} \to \mathbb{P}^5$. We will think of a point in $\mathcal{C}$ as a pair $(p,C)$ where $C$ is a plane conic and $p$ is a point on it.
In $\mathbb{P}^2$, we fix a point $q$ and a line $\ell$. Then the set of points $(p,C) \in \mathcal{C}$ with $q \in C$ forms a Weil divisor $\mathrm{D}_q$ on $\mathcal{C}$. Similarly, the set of points $(p,C) \in \mathcal{C}$ with $p\in \ell$ forms a Weil divisor $\mathrm{D}_\ell$.
Finally, let $K$ denote a fibre of the map $\pi$ over some point. In problem 18.4.O, Vakil uses the notation $\mathrm{D}_q \cdot K$ and $\mathrm{D}_\ell \cdot K$. I'm not sure what this notation means. My first guess is that it refers to intersection number, but intersection theory hasn't been introduced yet (we'll have to wait until Chapter 20 to see intersection theory), and there's no indication that one should look ahead to do this exercise. (edit: upon rereading, it seems the following is incorrect.) What's worse, it doesn't seem like the intersection theory that is introduced later is set up in a way that we could make use of here. (Chapter 20 only discusses the special case of an intersection of a sheaf with a collection of invertible sheaves.)
Am I mistaken in assuming this is related to intersection theory?