Upper and lower bounds for the norm of a linear combination of
vectors are given. Applications in obtaining various inequalities for the quantities
||x/||x|| - y/||y|| || and ||x/||y|| - y/||x|| ||, where x and y are nonzero
vectors, that are related to the Massera-Schäffer and the Dunkl-Williams
inequalities are also provided. Some bounds for the unweighted Čebyšev functional
are given as well.