B737 FLT CREW IS REQUIRED TO PERFORM AN EVASIVE MANEUVER DUE TO LACK OF ACFT SEPARATION DURING INITIAL APCH TO BWI.

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WHILE ON A RADAR VECTOR 'DOWNWIND' FOR RWY 10 IN IMC CONDITIONS; HEADING WAS APPROX 270 DEGS. WE WERE ABEAM THE FIELD; FLAPS 5 DEGS; SLOWED TO 180 KTS. THEN; POTOMAC APCH GAVE US HEADING 350 DEGS. WE SLOWED TO FLAPS 10 DEGS; ABOUT 170-160 KTS. WE WERE THEN ISSUED HEADING OF 020 DEGS. THE FO AND I NOTICED THE CTLR GETTING A COMPANY FLT NUMBER AND ANOTHER CARRIER FLT CONFUSED. THE FO MADE A COMMENT THAT WE WERE GOING TO BE VECTORED THROUGH THE LOC. I WAS WATCHING MULTIPLE TFC ON THE TCASII; BUT ONE OF CONCERN WAS JUST ON THE N SIDE OF THE LOC COMING STRAIGHT FOR US AT OUR ALT. I KEPT WATCHING THIS TFC AT OUR 12 O'CLOCK POS. I COULD NOT UNDERSTAND WHY OUR TCASII WAS NOT GIVING US AN RA. ATC WAS VERY BUSY AND WE COULD NOT GET AN INQUIRY TO ATC ABOUT THE TFC. I COULD NOT WAIT ANY LONGER. I REPLIED; 'WE'RE OUT OF4 HERE.' I DISCONNECTED THE AUTOPLT AND SHOVED THE THROTTLES FORWARD INITIATING A CLB AND STARTED A R TURN (TFC ON THE NOSE AND JUST SLIGHTLY L). THE OTHER CARRIER FLT INQUIRED ABOUT TFC AT THEIR 12 O'CLOCK POS. I CLBED FROM 3000 FT TO 3300 FT ABRUPTLY AND TURNED 30-50 DEGS R. I WAS AFRAID TO CLB ANY HIGHER BECAUSE THERE WERE OTHER ACFT IN CLOSE PROX. A BRIEF SECOND WENT BY WONDERING IF WE WERE GOING TO HIT THE TFC. AT THIS TIME; I DID NOT THINK THE TCASII PERFORMED THE WAY I THOUGHT IT SHOULD. THE TFC WENT FROM WHITE TO YELLOW AT OUR ALT; BUT NO AURAL WARNING TO RESPOND. I JUST COULD NOT SIT THERE AND KEEP WATCHING THE TFC GETTING CLOSER AND CLOSER. WHILE CLBING FROM 3000 FT TO 3300 FT; I WAS WATCHING THE TFC GETTING 300 FT VERT SEPARATION. FO THOUGHT HE SAW TFC PASS VERY CLOSE! ABOUT THAT TIME; ATC (NEW VOICE) SAID; 'COMPANY FLT NUMBER; TURN L HEADING 335 DEGS.' FO READ BACK FLT NUMBER; TURNING L TO HEADING 335 DEGS. I WAS HESITANT AT FIRST TO TURN L BECAUSE TFC WAS SLIGHTLY TO OUR L ON TCASII AND AT THIS POINT; I DID NOT HAVE ANY TRUST IN ATC. I SLOWLY STARTED A L TURN TO 335 DEGS. WE RECEIVED FURTHER VECTORS (L TFC PATTERN) TO JOIN LOC FOR RWY 10 AND CLRED FOR APCH. WHEN APCH HANDED US OVER TO BWI TWR; FO READ BACK GOING TO TWR AND ADDED; 'THAT WAS CLOSE.' THE APCH CTLR REPLIED; 'I KNOW.' FLT CONTINUED WITHOUT ANY FURTHER INCIDENT. THROUGHOUT THE WHOLE APCH; FO WAS CONTINUALLY 'IN THE LOOP' IN FACILITATING ME WITH SITUATIONAL AWARENESS AND CREW COORD.