Pressure Drop in a Pipe Calculation

Pressure is force per unit area. You also know it by PV = RT.

It is not sure that you get a vacuum. According to Bernouli's law in a flow the sum of static and dynamic pressures is constant. In your case due to the resistance of the pipe from B to atmosphere it could happen that static pressure at B is over the atmospheric pressure and thus no vacuum.

In an ejector vacuum occurs after the nozzle where the pressure is the outlet pressure and the fluid has its maximal velocity thus its smallest static pressure.

If you compare with your sketch B is in front of the "nozzle".

If you make air flow through B and air flows in same direction then pressure near A will drop down. This technique we use for shucking gases by forcing air and creating low pressure behind due kinetic energy.

If you make air flow through A and it flows in the same direction, then also air from B will be shucked in but not to the level it will do in case before.

Mathematically one need to workout all developed pressure at various points and flow achieved due to pressure drops. If you forcefully stop the flow, then things will be very different and that says about the pressure distribution in the duct.

Acording to the sketch, both ends of the large pipe are open. If this is true it changes things. Also some dimensions would be nice.

You'll need to say if this is true and how high the exhaust stack is, if it is closed on the "bottom".

Or is it horizontal, as drawn?

In fact both ends of the pipe can be open in free air even then the injected gas through inlet A or B can make flow only towards one side of the pipe and both ends of the pipe will also see pressure difference and flow. Air flow in any pipe will cause air flow in all pipes. This makes pressure changes in all pipes. Any pipes end having closed is as good as not being present in the duct or pipe assembly.

I have used one with similar design having a pipe and only one injector which also drags air from free end along with flow. Such forced flow using injected air can be seen in many applications.