  We investigate an extension of the Generalized Uncertainty Principle (GUP) in
three dimensions by modifying the three dimensional position and momentum
operators in a manner that remains coordinate-independent and retains as much
of the standard position-momentum commutators as possible. Moreover, we bound
the physical momentum which leads to an effective minimal length in every
coordinate direction. The physical consequences of these modified operators are
explored in two scenarios: (i) when a spherically-symmetric wave function is
`compressed' into the smallest possible volume; (ii) when the momentum is
directed in a single direction. In case (ii), we find that the three
dimensional GUP exhibits interesting phenomena that do not occur in one
dimension: the minimal distance in the direction parallel to a particle's
momentum is different from the minimal distance in the orthogonal directions.
