B757 FLT CREW INBOUND TO BWI AT 6000 FT EXPERIENCED CONFLICT WITH SAME ALT TFC AS CTLR ISSUED TURN AND TCASII ACTIVATE SIMULTANEOUSLY.

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I WAS CAPT ON FLT SCHEDULED ARRIVING BWI AT XA53 JUL/XA/05. WE WERE FLYING THE OTT6 ARR IN CLOUDY/LAYERED SKIES WITH CUMULO NIMBUS AROUND. POTOMAC APCH CTL GAVE US A DSCNT CLRNC TO CROSS DEALE INTXN AT 6000 FT. THE STAR CONTINUES AFTER DEALE INTXN TO RAVNN INTXN THEN A L TURN TO RIPKN INTXN AFTER WHICH; EXPECT RADAR VECTORS. AFTER WE WERE IN THE L TURN TO RIPKN INTXN; THE CTLR; I BELIEVE; ASKED OUR HDG; AND WE REPLIED WE WERE TURNING TOWARDS RIPKN INTXN. HE SAID 'TURN R TO 090 DEGS AND DSND TO 4000 FT.' THEN ALMOST IMMEDIATELY HE SAID 'TURN L TO 320 DEGS.' ALMOST SIMULTANEOUSLY; WE GOT A TCASII 'TFC' AND AN RA TO DSND. WE SAW B737 TFC AT APPROX 2 O'CLOCK POS; LEVEL; ABOUT 1 MI. WE CONTINUED L TO 320 DEGS AND DSNDED TO 4000 FT MSL. THE REST OF THE RADAR VECTORING WENT FINE; AND THE CTLR MENTIONED HE 'THOUGHT' HE HAD GIVEN US A 090 DEG HDG AFTER DEALE INTXN; WHICH NEITHER MYSELF NOR FO HEARD. AS WE SWITCHED CTLRS; THE ONE DURING THIS TIME APOLOGIZED AND SAID IT HAD BEEN A LONG DAY. I ASKED IF WE NEEDED TO DO PAPERWORK; AND HE SAID NO; BUT I THOUGHT I WOULD SUBMIT THIS JUST IN CASE.