You can try using one of the legs of the component cable, switching it over to composite video, then running composite video to the projector. Do this for all three cables - red, green, and blue.
FYI: The manual doesn't indicate it as 'automatic' it indicates that you have to pick what type of viewing you are doing. So, I would first double check the settings in the menu to make sure your 'application' is set properly.
Very few projectors I have used correctly switch between VGA-PC and VGA-Component video. Since you want VGA-Component, I would be sure to check that first.