Help setting up 360 with HD tv using component cable

Ok, I've just got an HD tv, which I'm now trying to set up my xbox with the component cable that the console came with. Taking off the scart adapter I had for my old tv, and plugging the those cables in the correct slots on the tv shows up picture and sound fine. I've had problems when changing the switch on the cable from TV to HDTV, but seem to have gotten 1080i on now. When I try to put it to 1080p, it just gives me a black screen. Might I be right in thinking I need to buy an HDMI cable to display 1080p from my console? The TV definitely supports either, but maybe doesn't pick it up when using the component?

Better double check to make sure that your tv is really 1080p compatible...because if it shows a picture for everything but that...usually means it doesn't display it. Difference between 720 to 1080 isn't so mind blowing that it is necessary anyways....so I would just stick with 720, but that is just me.