  The unzipping of vortex lines using magnetic-force microscopy from extended
defects is studied theoretically. We study both the unzipping isolated vortex
from common defects, such as columnar pins and twin-planes, and the unzipping
of a vortex from a plane in the presence of other vortices. We show, using
analytic and numerical methods, that the universal properties of the unzipping
transition of a single vortex depend only on the dimensionality of the defect
in the presence and absence of disorder. For the unzipping of a vortex from a
plane populated with many vortices is shown to be very sensitive to the
properties of the vortices in the two-dimensional plane. In particular such
unzipping experiments can be used to measure the ``Luttinger liquid parameter''
of the vortices in the plane. In addition we suggest a method for measuring the
line tension of the vortex directly using the experiments.
