Join Date: Apr 2016
Location: Twin Cities, MN
Year/Make/Model: 2001 Vulcan 800 Classic
Pretty sure GFISH is correct.
It looks like one of the 3 hoses that connect to the bottom of the air filter assembly.
When you reassemble everything, it'll come in under the carburetor and plug into the back of the black plastic assembly, along with the larger Air Switch Valve hose and a smaller hose that I don't know the name of. Figuring out which hose goes on which nozzle is pretty easy. There are 3 of each, and they're all different sizes. The nozzle for that hose sticks out further than the other two.
I've become pretty familiar with this area of my back over the last weekend, as trying to get those 3 hoses reconnected is the only thing stopping my bike from being on the road right now...
I hope you have better luck.
2001 Vulcan 800 Classic