  Measuring the full three-dimensional motions of extra-galactic objects in the
Universe presents a seemingly insurmountable challenge. In this paper we
investigate the application of a technique to measure tangential motion that
has previously only been applied nearby within the Local Group of galaxies, to
clusters of galaxies far beyond its borders. We show that mapping the mean
line-of-sight motion throughout a galaxy cluster could in principle be used to
detect the "perspective rotation" induced by the projection of the cluster's
tangential motion into the line-of-sight. The signal will be most prominent for
clusters of the largest angular extent, most symmetric intrinsic velocity
distribution and surveyed with the largest number of pointings possible. We
investigate the feasibility of detecting this signal using three different
approaches: measuring line-of-sight motions of individual cluster members;
taking spectra of intracluster gas; and mapping distortions of the Cosmic
Microwave Background radiation. We conclude that future spectroscopic surveys
of 1000's of members of nearby galaxy clusters hold the most promise of
measuring cluster tangential motions using this technique.
