We do stuff like this for toilets all the time, but the blue joists would run through, be doubled, and hang off the joists on the sides, then you would hang the trimmed joist off the hanger.

You would also probably want to block off between joists above the wall which will help the joist that continues beyond the wall to stay upright and help take the weight of whatever is above the wall.

Make sure that there is no structure above the piece of the joist you are taking out.

Re-route that electrical wire to the other side of the joist so that it doesn’t get in the way of the hanger and header.

If that sounds beyond your ability the get a professional to do it. And since they will cost as much as a new projector, then just buy a new projector that throws the distance that you need it to.

Before any of that: read your projectors manual in case you’re missing something that will allow you to adjust the projection to the space you have.