What causes vibration in rear when applying brakes

It sounds like your brake rotors need to be replaced, they could be warped from overheating. This could be caused by applying the brakes too much while going downhill.

This is kind of a long shot, but you've done all the logical things already. You didn't state what make, model, year so I don't know if this will apply or not. I had a 2002 Intrepid that did this (infrequently at first, but became more pronounced) when I was turning slightly while braking. It turned out to be the inner tie rod ends, where they connect to the steering rack and pinion. The bushings were badly worn, and you could feel this with the wheels off the ground.

you might have a bad front rotor ...warped or burnt ...I would check there 1st

UNLESS there is a major front end issue the most common cause of the vibration is the rotors being out of round. this is easily correct by having a brake shop resurface the rotors. also the vibration could be coming from the rear as well. the way to determine if it rear or front is the next time you brake somewhere that it usually has the vibration try applying only the parking brake instead of the brake pedal. NOTE....when making this check be sure to hold the parking brake release button to prevent lock up of the brakes. the parking brake only works the rear brakes. this will determine where it is coming from. the back brakes will have either brake drums or either brake rotors. same philosophy but different operating mechanics. either way it can all be corrected by resurfacing the rotors and/or drums

If it has drum brakes get the drums machined or if it has discs get the discs machined to stop the vibration as with age and heat and water splashes over time the discs and or drums go out of round causing the pads(discs) shoes (drum brakes)to cause a vibration when they are applied,

chack the transmission shafts

do u feel the vibration in the steering wheel or in the seat,i bet the most vibration was coming out of the front brakes and a little out of the rear brakes,u might want to change the rear brakes if the vibration is still annoying,also were the front wheels torqued to spec when installed?? if not u might end up with warped front brakes again,hope this helped

Check your front and rear caliper slides for sticking,it sounds like one may be stuck.

If the brakes are fine, try rotating the wheels.
The tires it self could have a bad belt in it and have a soft spot. Try putting the front tires to the rear and the rear to the front, double check the tire pressure and see if the vibration go's away.
If the vibration is still there then the tires are good and it's the brake rotor or worn steering parts and or needs a wheel alignment. As the front end drops the toe-in changes geometry and may just be out enough to cause the wheels to vibrate.
Good luck and hope this help. I fine the brake rotors inexpensive enough at a parts store that it's better to change them rater then have then turned to a smooth surface again. If your handy try changing the rotors with new one's. Auto Zone has the parts and would give the step by step instructions to do the job.

NEED NEW ROTORS