The answer isn't as simple as what linear logic dictates - "vapor is not smoke, therefore it can't". On the contrary, vaping can set off smoke detectors. For an electronic cigarette to set off a smoke detector, there are certain variables that come into play, such as the type of smoke detector and proximity to the smoke detector. The two most common smoke detectors are based on photoelectric and ionization principles. Photoelectric smoke detectors use unbroken beams of light that, when interrupted, cause the shrill shrieking we all know. Ionization based detectors are different in the sense they are built to detect particle density in the air.
Consider a photoelectric detector. If you blow a thick cloud, thick enough to interrupt the light beam, the alarm will go off - no two ways about it. The same goes for ionization based detectors because depending on the e-liquid mix you use, you could be creating vapor that is denser than water vapor, which will again set the alarm off. Both these views are consistent with what vapers say about this in Google - some say "yes, I have set my smoke detector off while vaping", some say "no". Their contrasting experiences are the result of the variables we just mentioned, so it could be a good idea to keep these in mind when vaping around a smoke detector. There is also the matter of being too close to a detector when you're puffing away on your ecig. Obviously, the closer you are to a smoke detector, the higher the chances of setting it off so exercise some caution and don't risk it when you really don't need to. Is it really worth being put on the no-fly list just because you had to vape on the plane? Probably not.