DA40 INSTRUCTOR AND STUDENT LOSE CONTACT WITH APCH CTL DURING PRACTICE IFR FLT.

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DURING AN INST TRAINING FLT; MY STUDENT AND I WERE ON VECTORS FOR THE ILS APCH INTO BFI RWY 31L. WE WERE LISTENING TO THE RADIO DURING THE VECTORS AS INSTRUCTED AND THE FLT WAS PROGRESSING AS EXPECTED. THE VECTORS STARTED BECOMING UNUSUAL AND WE STARTED TO WONDER WHY WE WERE GETTING VECTORED SO FAR AWAY FROM THE ARPT ONTO THE LOC. BEING THAT THE SVC VOLUME FOR A LOC IS ONLY 10 DEGS AT 18 NM AWAY WE WONDERED WHY WE WERE AT A 90 DEG INTERCEPT TO THE LOC ABOUT 22 NM AWAY. ABOUT THE TIME WE WERE GOING TO CALL ATC (SEATTLE APCH) AND ASK WHAT WAS GOING ON; THEY CALLED US. WE HEARD OUR CALL SIGN AND IF WE WERE STILL ON FREQ. WE RESPONDED THAT WE WERE; AND THEY RESPONDED THAT THEY HAD BEEN TRYING TO GET A HOLD OF US FOR QUITE SOME TIME AND THAT WE ALMOST ENTERED A MIL HOT ZONE. NEITHER MY STUDENT NOR I EVER HEARD OUR CALL SIGN OTHER THAN THE REQUEST IF WE WERE ON THE FREQ. AT THAT POINT THEY SUGGESTED WE MONITOR GUARD FREQ IN THE FUTURE AND ULTIMATELY HANDED US OFF TO ANOTHER SEATTLE APCH FREQ. IT SEEMED THAT WE MISSED A HDOF TO ANOTHER FREQ DURING THE FLT AND WHEN WE NEVER RPTED ON THE NEW FREQ; SEATTLE APCH BEGAN TO LOOK FOR US. NORMALLY; IF A PLT DOESN'T RESPOND TO A RPT; ATC WILL TRY ANOTHER ATTEMPT UNTIL RECEIVING CONFIRMATION. WE NEVER MADE CONFIRMATION OF THE INSTRUCTIONS AND WE BELIEVE THAT ATC ASSUMED THAT WE HEARD THEM. WE WERE FLYING TOWARDS THE MOUNTAINS ON VECTORS WITHOUT ANY FORM OF ATC CTLING US FOR ROUGHLY 10 MINS. WE WERE FORTUNATE THAT WE WERE IN VMC TO IDENT THE MOUNTAINS IN CASE WE GOT VECTORED TOO CLOSE. THIS COULD HAVE RESULTED IN CFIT DUE TO THE MISCOM OR BREAKAGE OF THE LINE OF COM BTWN APCH CTL AND US. CONFIRMATION THAT INFO IS RECEIVED IS VITAL.