  We propose a novel representation of cosmic microwave anisotropy maps, where
each multipole order l is represented by l unit vectors pointing in directions
on the sky and an overall magnitude. These "multipole vectors and scalars"
transform as vectors under rotations. Like the usual spherical harmonics,
multipole vectors form an irreducible representation of the proper rotation
group SO(3). However, they are related to the familiar spherical harmonic
coefficients, alm, in a nonlinear way, and are therefore sensitive to different
aspects of the CMB anisotropy. Nevertheless, it is straightforward to determine
the multipole vectors for a given CMB map and we present an algorithm to
compute them. Using the WMAP full-sky maps, we perform several tests of the
hypothesis that the CMB anisotropy is statistically isotropic and Gaussian
random. We find that the result from comparing the oriented area of planes
defined by these vectors between multipole pairs 2<=l1!=l2<=8 is inconsistent
with the isotropic Gaussian hypothesis at the 99.4% level for the ILC map and
at 98.9% level for the cleaned map of Tegmark et al. A particular correlation
is suggested between the l=3 and l=8 multipoles, as well as several other
pairs. This effect is entirely different from the now familiar planarity and
alignment of the quadrupole and octupole: while the aforementioned is fairly
unlikely, the multipole vectors indicate correlations not expected in Gaussian
random skies that make them unusually likely. The result persists after
accounting for pixel noise and after assuming a residual 10% dust contamination
in the cleaned WMAP map. While the definitive analysis of these results will
require more work, we hope that multipole vectors will become a valuable tool
for various cosmological tests, in particular those of cosmic isotropy.
