NMAC IN CLASS B AIRSPACE. UNAUTH PENETRATION OF AIRSPACE.

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ON DEPARTING RWY 12 AT VGT (NORTH LAS VEGAS); TWR TOLD US TO TURN R HDG 220 DEGS AND NOT TO CALL APCH UNTIL ADVISED (WE HAD ALREADY BEEN GIVEN A SQUAWK CODE BY GND CTL). TO AVOID CLASS B I TOLD MY STUDENT TO TURN FURTHER R TO 240 DEGS. TWR APPARENTLY FORGOT US UNTIL WE CALLED CLR OF THEIR AIRSPACE. THEY THEN SAID; 'OH; YEAH; CONTACT APCH.' UPON CONTACTING APCH WE WERE TOLD; 'NEXT TIME CONTACT US PRIOR TO ENTERING BRAVO AIRSPACE. TURN L HDG 090 DEGS (MAINTAIN 5500 FT).' APPARENTLY WE HAD DRIFTED FURTHER S THAN I HAD THOUGHT. I WAS UNFAMILIAR WITH THE AREA OVER THE CITY. WHILE THIS WAS GOING ON MY PRIMARY (PRE-PVT PLT) STUDENT HEARD THE ALT RESTRICTION BUT I DID NOT. WHEN HE TRIED TO LEVEL OFF I TOLD HIM TO KEEP CLBING. SHORTLY AFTER I HEARD ATC ADVISE A DEPARTING BOEING 727 TO LOOK FOR A CESSNA 172. I TURNED AND SAW HIM LOW AT OUR 4-5 O'CLOCK POS IN A CLBING R TURN HEADED DIRECTLY FOR US. IT QUICKLY BECAME APPARENT THAT WE WERE ON A COLLISION COURSE AND I TOOK THE CTLS FROM MY STUDENT WHO WAS DUTIFULLY CONTINUING TO CLB LIKE I HAD TOLD HIM. AFTER THE B727 PASSED DIRECTLY OVERHEAD; AND I BRACED FOR THE WAKE TURB JOLT; I HEARD THE B727 PLT SAY OVER THE RADIO; 'I'VE NEVER SEEN A CESSNA DIVE LIKE THAT BEFORE!' CONTRIBUTING FACTORS: INSTRUCTOR FATIGUE; UNFAMILIARITY WITH THE AREA; EXPECTATION; AND TASK SATURATION; AND BLINDLY FOLLOWING AN ASSIGNED HDG. CORRECTIVE ACTIONS: BETTER PREFLT PREPARATION BY THE CFI MIGHT HAVE PREVENTED THE INCURSION. ALSO; FATIGUE (LACK OF SLEEP AND LONG HRS) CONTRIBUTED A GREAT DEAL TO THE MISTAKES -- BETTER REST. A LITTLE ASSERTIVENESS BY THE STUDENT (WHO NOW KNOWS CFI'S MAKE MISTAKES; TOO) MIGHT HAVE PREVENTED THE LOSS OF SEPARATION.