I'm thinking out loud here so bear with me but -
You are going from an F output on the VCR to an RCA (Composite) input on the projector?
If so the signals are not the same (Multichannel RF vs Composite Video), if you go from the VCR Video Out (RCA - not F) to the projector then this should work. I utilize this setup for my tuner for my system. You'll need an RCA-RCA copper/copper coax. RG6 CATV may work but it is not correct for the signal in question and you'll get better performance with the copper/copper coax. Distance will not be an issue.
Without seeing anything this is an assumption, if I am off base ignore it.