Multipole vectors of completely random microwave skies for $l \leq 50$
  The statistical cosmological principle states that observables on the
celestial sphere are sampled from a rotationally invariant distribution.
Previously certain large scale anomalies which violate this principle have been
found, for example an alignment of the lowest multipoles with the cosmic dipole
direction. In this work we continue the search for possible anomalies using
multipole vectors which represent a convenient tool for this purpose. In order
to study the statistical behavior of multipole vectors, we revisit several
construction methods. We investigate all four full-sky foreground-cleaned maps
from the Planck 2015 release with respect to four meaningful physical
directions using computationally cheap statistics that have a simple geometric
interpretation. We find that the full-sky SEVEM map deviates from all the other
cleaned maps, as it shows a strong correlation with the Galactic Pole and
Galactic Center. The other three maps COMMANDER, NILC and SMICA show a
consistent behavior. On the largest angular scales, $l \leq 5$, as well as on
intermediate scales, $l = 20, 21, 22, 23, 24$, all of them are unusually
correlated with the cosmic dipole direction. These scales coincide with the
scales on which the angular power spectrum deviates from the Planck 2015
best-fit {\Lambda}CDM model. In the range $2 \leq l \leq 50$ as a whole there
is no unusual behavior visible globally. We do not find abnormal intramultipole
correlation, i.e. correlation of multipole vectors inside a given multipole
without reference to any outer direction.
