  We use three different cosmological dark matter simulations to study how the
orientation of the angular momentum vector (AM) in dark matter haloes evolve
with time. We find that haloes in this kind of simulations are constantly
affected by a spurious change of mass, which translates into an artificial
change in the orientation of the AM. After removing the haloes affected by
artificial mass change, we found that the change in the orientation of the AM
vector is correlated with time. The change in its angle and direction (i.e. the
angle subtended by the AM vector in two consecutive timesteps) that affect the
AM vector has a dependence on the change of mass that affects a halo, the time
elapsed in which the change of mass occurs and the halo mass. We create a
Monte-Carlo simulation that reproduces the change of angle and direction of the
AM vector. We reproduce the angular separation of the AM vector since a look
back time of 8.5 Gyrs to today ( $\rm \alpha$) with an accuracy of
approximately 0.05 in $\rm cos(\alpha)$. We are releasing this Monte-Carlo
simulation together with this publication. We also create a Monte Carlo
simulation that reproduces the change of the AM modulus. We find that haloes in
denser environments display the most dramatic evolution in their AM direction,
as well as haloes with a lower specific AM modulus. These relations could be
used to improve the way we follow the AM vector in low-resolution simulations.
