So to do this properly you will need to head out the joists which means where your blue marks are you would run a double floor joist from joist to joist and “heading” off the joist you want to notch you would then do the same thing on the other side creating your desired space for your projector. Everything would need joist hangers. I will add the fact that you are asking this question means that you should get someone who knows what they are doing to at the very least come look at this because I can tell you the proper way to frame this but there can be any number of reasons you shouldn’t do this and I have a single picture to go off.