A B727-200 FLC RECEIVES VECTORS FOR ARR INTO DCA THAT APPEARS TO CONFUSE THE CTLR. FLT LATER PENETRATES THE P-56 SECTION N OF DCA; DC.

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APCHING DCA AT 10000 FT MSL AND 250 KIAS FROM THE BILTT INTXN (ATR 275 DEGS/40 DME). WE RECEIVED A VECTOR OF 320 DEGS FROM WASHINGTON APCH. AFTER A FEW MINS THE CTLR ASKED US WHAT OUR HEADING WAS. WE RESPONDED THAT WE WERE 320 DEG HDG ASSIGNED. HE SAID SOMETHING LIKE; 'IT DOESN'T LOOK LIKE THAT DOWN HERE.' WE XCHKED OUR HEADING INDICATORS WITH THE MAGNETIC COMPASS AND CONFIRMED THEY WERE THE SAME AND ASKED WHAT HEADING HE WOULD LIKE. WE RECEIVED NO RESPONSE. THE CTLR THEN QUERIED ANOTHER ACFT ABOUT THE WIND AT ALT. SHORTLY AFTER WE WERE GIVEN A R TURN TO 020 DEGS. FROM THEN ON WE RECEIVED VECTORS TO INTERCEPT THE RIVER VISUAL TO RWY 19 AT DCA. SUPPLEMENTAL INFO FROM ACN 475174: THE CTLR REPLIED THAT HE COULDN'T UNDERSTAND OUR POS IF WE WERE FLYING 320 DEGS. PROBABLY 20-40 SECONDS AFTER THE CTLR'S PREVIOUS XMISSION; THE CTLR TOLD US TO TURN R TO 020 DEGS. ABOUT 1 HR AFTER THE INCIDENT THE CREW WAS INFORMED THAT WE HAD TRANSGRESSED AREA P-56 (WHITE HOUSE). ONCE THE CTLR SAW THE IMPENDING SIT; HE SHOULD HAVE CLRLY AND FORCEFULLY GIVEN US TURN INSTRUCTIONS WITHOUT MULLING OVER WHETHER WE WERE ACTUALLY FLYING THE ASSIGNED HEADING.