AN MD80 FLT CREW AND APCH CTLR HAVE CONFUSION ABOUT WHAT THE ASSIGNED HEADING WAS.

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UPON BEING HANDED OFF TO DULLES (IAD) APCH CTL WE WERE GIVEN A CLRNC TO DEPART BARIN INTXN (BRU 0070/15 DME) ON A 090 DEG HDG FOR APCH TO RWY 19R. I REPEATED BACK CLRNC AND WROTE DOWN 090 DEGS FROM BARIN ON NOTEPAD. UPON REACHING BARIN THE CAPT (PF) TURNED TO THE 090 DEG HDG. ABOUT 2-3 MINS PASSED AND CTLR TOLD US TO TURN L TO A 350 DEG HDG AND INDICATED THAT WE TURNED TO 090 DEGS IN ERROR. AFTER A FEW MINS; CTLR ADVISED US WE WERE GIVEN A 010 DEG HDG OFF OFF BARIN INTXN. I ADVISED HIM THAT I THOUGHT HE GAVE US A 090 DEG HDG OFF BARIN; AND THAT I REPLIED WITH A READBACK OF 090 DEGS. I DON'T BELIEVE THERE WAS A CONFLICT AND NOTHING FURTHER WAS SAID; AND I DON'T KNOW IF ANY FURTHER ACTION WILL BE TAKEN. THIS WAS OUR FIRST TIME INTO DULLES IN MANY YRS; SO WE WERE NOT FAMILIAR WITH COMMON VECTORS FOR VARIOUS ARR FIXES. PERHAPS IF WE WOULD HAVE EVALUATED THE DIRECTION OF THE VECTOR WITH THE RWY CONFIGN WE WOULD HAVE QUESTIONED IT. THIS WAS OUR 5TH LEG OF THE DAY AND FATIGUE MAY HAVE PLAYED SOME PART IN THIS.