  A powerful tool of investigation of Fano varieties is provided by exceptional
collections in their derived categories. Proving the fullness of such a
collection is generally a nontrvial problem, usually solved on a case-by-case
basis, with the aid of a deep understanding of the underlying geometry.
Likewise, when an exceptional collection is not full, it is not straightforward
to determine whether its "residual" category, i.e., its right orthogonal, is
the derived category of a variety. We show how one can use the existence of
Bridgeland stability condition these residual categories (when they exist) to
address these problems. We examine a simple case in detail: the quadric
threefold $Q_3$ in $\mathbb{P}^{4}$. We also give an indication how a variety
of other classical results could be justified or re-discovered via this
technique., e.g., the commutativity of the Kuznetsov component of the Fano
threefold $Y_4$.
