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I'm running a hessian matrix calculation(ibrion=5) for a big system. The calculation takes a long time, and I wonder how I can continue the calculation in case of a machine crash. Does vasp save the intermediate calculations of the hessian matrix? Thank you very much.

no, VASP does not save the intermediate steps. in case of a crash you have to start all over again. Please note that
--) use IBRION = 5 only for post-processing a converged run
--) by choosing seective dynamics, you can pick out the modes which are reasonable (decreasing the number of vibrational degrees of freedom will speed up the run significantly)

After the calculation, VASP outputs the eigenvectors and eigenvalues of the dynamical matrix(at gamma point, I think) in OUTCAR. This output is divided into 6 columns:x, y, z, dx, dy and dz. I believe the x, y, z are the components of the eigenvectors. But I have no idea what are the dx, dy and dz. What are these 3 columns correspond to? Thank you.

no, x,y and z are the positions of the ions (as read from POSCAR), dx, dy, and dz are the eigenvectors .

If you are interested, I wrote myself an extension to finite_diff.F and main.F
It's somehow alpha-stage, it does not work always, but most of the time. And it's tricky, if you have more than one restart to do.

VASP shows both mass weighted and non-mass weighted values for the eigenvectors (dx,dy,dz). what are the units for the non-mass weighted eigenvectors? are they in angstrom? i did frequency calculations for water, and VASP calculated non-mass weighted eigenvectors of magnitude 0.7. this seems very high compared to the 0.96 angstrom bond length of h-o in water.

nvm, just realized that eigenvectors should be dimensionless