A CRJ REGIONAL JET FLC ACCEPTS A TURN FOR ANOTHER ACR FLT AND IS REASSIGNED A HEADING BY DEP CTLR AT DTW; MI.

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I WAS THE PNF FOR FLT FROM DTW TO CVG. A DC9 WAS CLRED FOR DEP FROM RWY 3C. PRIOR TO THAT WE WERE HELD FOR TKOF FOR ANOTHER ACFT TO LAND ON RWY 3C. AFTER THAT ACFT LNDG WE WERE CLRED FOR TKOF ON RWY 3C WITH THE HEADING OF 060 DEGS. UPON 2000 FT AND A HEADING OF 060 DEGS; DTW ATC RESPONDED 'ACR X TURN R TO A HEADING OF 210 DEGS.' IN RESPONSE TO ATC INSTRUCTIONS; I READ BACK INSTRUCTIONS; 'ACR X TURN R TO A HEADING OF 210 DEGS.' I TURNED THE HEADING BUG FOR THE PF TO A HEADING OF 210 DEGS. APPROX 2 MINS LATER; ATC ASKED ME WHAT HEADING WE WERE FLYING. I RESPONDED 'A HEADING OF 210 DEGS.' DTW ATC SAID 'NO; THAT WAS FOR ACR Y.' ATC RESPONDED WITH A DIFFERENT HEADING OF 'TURN L TO A HEADING OF 120 DEGS.' I REMARKED 'SIR; I RESPONDED BACK TO YOU A HEADING OF 210 DEGS WITHOUT A RESPONSE FROM YOU;' WE THEN WENT OFF TO THE HEADING OF 120 DEGS. 30 SECONDS LATER ATC TURNED US BACK TO A HEADING OF 210 DEGS; THEN HANDED US OFF TO ANOTHER CTLR. WHILE THIS EVENT TOOK PLACE; THE CAPT HAD THE TCASII SYS UP TO A 20 MI RANGE; AND THE ONLY ACFT WE COULD FIND WAS OFF TO AN E HEADING AROUND A 20 MI RANGE; NOT EVEN CLOSE TO A HEADING OF 210 DEGS.