To act vacuously is to do something in a mindless or blank way. Staring vacuously at your textbook is not the same as actually studying.
Vacuous things lack any real intelligence or thought — they're empty and a little bit stupid. Doing something vacuously reflects this same mindlessness. If someone orates vacuously, their words are hollow and meaningless. Smile vacuously at someone and they're sure to see that your heart's not really in it. The Latin root of this word means "empty, void, or free."