can anyone tell me if the hard shell backpack that comes with the phantom 3 will fit in an airplane's overhead compartment?  I will be traveling with my UAV a lot.

What about issues with bringing the batteries on planes?

Well for example Qatar Airways allows hand luggage of size 50x37x25cm and the Hardshell backpack is approx 31.8 cm wide x 44.7cm high x 22.9 cm deep so yeah you should be able to take it on a plan, just check your airlines restrictions.
As for batteries, no issue as hand luggage on most airlines.

You can now buy the official DJI Hardshell Backpack for the Phantom 3

Remember to buy extra propellers and one or two spare batteries to keep you flying

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