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.