A PIPER TOMAHAWK INSTRUCTOR PLT AND HIS STUDENT TURNED TO THE WRONG HDG AFTER TKOF. TRACK AND HDG DEV. ERROR ADMITTED. NMAC.

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WE WERE TAKING RWY 13R AT BRO; AND I WAS CLOSING THE DOOR OF OUR SINGLE ENG PIPER TOMAHAWK. MEANWHILE; ATCT GAVE US OUR DEP INSTRUCTIONS. I DID NOT WRITE IT DOWN (NOR DID MY STUDENT); BUT I ASKED HIM IF HE HEARD WHAT I HEARD (350 DEGS AT 2000 FT). I ALSO DID NOT CATCH HIS READBACK. I BELIEVE THAT OUR REAL INSTRUCTIONS WERE A R TURN TO 250 DEGS AT 2000 FT. WE DEPARTED AND STARTED OUR TURN TO 350 DEGS (L TURN) AS ATCT SWITCHED US TO VALLEY APCH CTL. VALLEY ASKED US TO VERIFY OUR CLBOUT INSTRUCTIONS AND I SAID 350 DEGS AT 2000 FT. HE SAID IT SHOULD HAVE BEEN 250 DEGS AT 2000 FT AND WE SHOULD TURN NOW TO 270 DEGS. WE CONTINUED OUR L TURN WHICH BROUGHT US BACK TOWARD THE ARPT. MEANWHILE A DC9 WAS DEPARTING ON RWY 13R BEHIND US. ATC IMMEDIATELY TOLD US TO TURN TO 180 DEGS FOR TFC. WE SAW THE TFC AND DID AVOID HIM; HOWEVER THE DC9 COMMENTED TO ATC THAT HE HAD A 'WINDSHIELD FULL OF [THE TOMAHAWK].' THIS SCENARIO CONFIRMS THE IMPORTANCE OF WRITING DOWN CLRNCS; MONITORING YOUR STUDENTS; AND ELIMINATING DISTRS (CLOSING THE DOOR) IN CRITICAL PHASES OF FLT. IT ALSO EMPHASIZES THE IMPORTANCE OF READBACKS AND VERIFICATION OF CLRNCS WHENEVER THERE IS A QUESTION (NO MATTER HOW BUSY ATC IS).