projector video?

Yes, using the component outputs from your cable box and putting them into your projector will work wonderfully. I do it myself, and people often comment on how cool it is. Also, you could put the component outputs into an A/V receiver and then into your projector so that you could use it for more then just TV (DVD for example). Just one note of caution: if your projector does not have component inputs, and you are using a component to VGA cable, be sure your projector can read 525I format as well as VGA and XVGA. otherwise, you will get no picture.

if your cable box has component outputs then you will be able to hook that up to the projector (and a stereo for sound)

if your stereo has video inputs and outputs then you can put it between the cable box and projector