Why are vacancy rate and unemployment rate negatively correlated? Why is this the case?

Since Vacancy rate is defined as following,

let $A,Q,U$ denote number of vacancies in the economy, labor force, unemployed respectively.

$$\frac{A}{A+Q-U}$$

Here we can see that if unemployed increase vacancy rate would go up?

Why is there a negatively correlation then? Take the beveridge curve as an example :

https://en.wikipedia.org/wiki/Beveridge_curve
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.