  We extend a class of recently derived thermodynamic uncertainty relations to
vector-valued observables. In contrast to the scalar-valued observables
examined previously, this multidimensional thermodynamic uncertainty relation
provides a natural way to study currents in high-dimensional systems and to
obtain relations between different observables. Our proof is based on the
generalized Cr{\'a}mer-Rao inequality, which we interpret as a relation between
physical observables and the Fisher information. This allows us to develop
high-dimensional versions of both the original, steady state uncertainty
relation and the more recently obtained generalized uncertainty relation for
time-periodic systems. We apply the multidimensional uncertainty relation to
obtain a new constraint on the performance of steady-state heat engines, which
is tighter than previous bounds and reveals the role of heat-work correlations.
As a second application, we show that the uncertainty relation is connected to
a bound on the differential mobility. As a result of this connection, we find
that a necessary condition for equality in the uncertainty relation is that the
system obeys the equilibrium fluctuation-dissipation relation.
