Vector Bundle definition compared to direct product

Question

I read that the simplest vector bundle is $E = M \times V$ where $M$ is a manifold and $V$ is some $r$-dimensional vector space. I suppose my confusion is with notation. My understanding is that if I say $A \times B$ it means pairs of elements $(a,b)$ with $a \in A$ and $b \in B$, but a vector bundle means for each point we have an entire space of vectors...which is not the same as one point paired with one vector. So how come they can write $E=M \times V$?