Well DP and HDMI are two completely different things, both physical and signal

. On the Mac the DP on that can do HDMI signals, but the one on the 7970 can't, hence why you need an active adapter to convert the DP signals to HDMI. A passive adapter that only rearranges the pins hence won't work as it does nothing to the signals.
Does your card not have a HDMI or DVI port? And if this is for a third monitor, what connections does it have? You can get active mini DP to DVI for far cheaper.