AN ACR B737-500 LINED UP ON THE S RWYS AT LAX. IT WAS CLRED TO THE N SIDE.

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ATC PROVIDED A VECTOR FOR BASE TO RWY 24R AT LAX FOR A VISUAL APCH TO FOLLOW A B737 TURNING FINAL. THE FIELD; INCLUDING BOTH RWY COMPLEXES; WAS IN SIGHT FROM DOWNWIND AND DURING THE INITIAL TURN TO BASE. I WAS 'PADLOCKED' TO THE B737 TFC AS THE HAZE BECAME A FACTOR WHILE LOOKING INTO THE SUN. THE ILS TO RWY 24R WAS OTS AND WE INDICATED FROM OUR FMS WE WERE SLIGHTLY S OF EXTENDED CTRLINE. WE WERE BOTH CONCERNED WITH 'SEQUENCING' OURSELVES REF OUR TFC AND WERE NOT MONITORING OUR INSTS ACCURATELY. AS WE ROLLED OUT ON FINAL; I MENTIONED A GND REF I HAD NOT PREVIOUSLY NOTICED. WE BOTH SIMULTANEOUSLY REALIZED OUR PROB AS LAX TWR CALLED TO SEND US TO THE PROPER SET OF RWYS. THE REQUIREMENT FOR PLTS TO PICK UP VISUAL CONTACT ON COMPETING ARRIVING TFC WHILE MANEUVERING TO FINAL IS A NECESSITY AT ALL BUSY STATIONS TO KEEP TFC FLOWING. IT IS CRITICAL FOR PLTS TO USE ALL MEANS AVAILABLE AND DIVIDE DUTIES TO ENSURE THERE IS NO TFC CONFLICT FIRST AND THEN TO NAV AS ACCURATELY AS POSSIBLE. WE ATTEMPTED TO DO BOTH; BUT FIXATED ON THE FIRST. WHILE THERE WAS NO SAFETY ISSUE DUE TO OUR CLRING; WE WERE IN A POS NOT ANTICIPATED BY ATC AND COULD HAVE BEEN A CONFLICT TO UNKNOWN TFC. FMC'S SHOULD BE MONITORE D AT THE SMALLEST SCALES APPROPRIATE AND OTHER AVAILABLE NAVAIDS USED. (WE BRIEFED ILS RWY 24L AS BACKUP TO RWY 24R; BUT DIDN'T USE IT; AS WE CONFIDENTLY CALLED VISUAL ONLY TO LOSE IT.) FURTHER; THERE SHOULD BE PROC AND NO 'RETRIBUTION' FOR BEING AN AIRFIELD IN THE SUN MOMENTARILY. OUR MANEUVER IS DONE ROUTINELY AT LAX WITH LITTLE PROB. WE SIMPLY OVERSHOT DUE TO PRIORITIZING TFC OVER RWY LINEUP AND THEN BELIEVING THE RWYS IN FRONT OF US WERE THE RIGHT ONES.