  Uncertainty principle is one of the cornerstones of quantum theory. In the
literature, there are two types of uncertainty relations, the operator form
concerning the variances of physical observables and the entropy form related
to entropic quantities. Both these forms are inequalities involving pairwise
observables, and are found to be nontrivial to incorporate multiple
observables. In this work we introduce a new form of uncertainty relation which
may give out complete trade-off relations for variances of observables in pure
and mixed quantum systems. Unlike the prevailing uncertainty relations, which
are either quantum state dependent or not directly measurable, our bounds for
variances of observables are quantum state independent and immune from the
"triviality" problem of having zero expectation values. Furthermore, the new
uncertainty relation may provide a geometric explanation for the reason why
there are limitations on the simultaneous determination of different
observables in $N$-dimensional Hilbert space.
