A "vector pointing at Alpha Centauri" is an incomplete definition, since a vector has a magnitude and direction. Also, Alpha Centauri has a declination of –60° 50′ 02.3737″, which makes it necessary to adjust for the time of the day. That would make people dizzy, which is unfortunate.

So I propose the following very simple definition:

Up - The direction defined by the position vector of Polaris with reference to the center of the Earth (not to be confused with North).

Down - The direction opposite of Up (See also: South).