Why are vacancy rate and unemployment rate negatively correlated?
Adopting your notation, the vacancy rate at any given time is defined as $A/Q$. There is no mechanical relationship between the unemployment rate $U/Q$ and vacancy rate (A/Q).

They are negatively related because when vacancy rate is high, it means there are many unfilled job openings, and they are unfilled probably because most people already have a job, thus lower unemployment rate. And vice versa.