I've worked in construction for 20+ years and have had to get engineered fixes to reframe around HVAC and/or plumbing in similar situations.

You're best bet is to double up your existing floor joists to the right and left of the intended joist to be cut. These need to run full length from one bearing point to the other. Buy yourself some 3 inch and 1-1/2 inch wide joist hangers from a nearby home improvement store, 4 of the 3 inch wide and 2 of the 1-1/2 wide hangers. Looks like 2x10 joists so make sure the hangers are at least 8 inches tall. Temporarily support the joist to be cut with a 2x4 down to the floor. Cut your intended joist as needed to create the pocket for the projector. Place a 2 ply 2x10 (or 2x8, it needs to be the same height as existing joist) across each end of the cut joist in-between the exciting joists so that everything is perpendicular and square. Temporarily attach with screws or nails. Attach one 3 inch wide hanger to each existing doubled up joist with 2-1/2 inch joist hanger nails or joist hanger screw. Do not use any other kind of nail or screw with these hangers. Finally attach the 1-1/2 inch wide hangers around the joist that was cut to the side of the new 2 ply joist header, again using joist hanger nails or joist hanger screws.