  We give a criterion for a projectivized toric vector bundle to be a Mori
dream space and describe its Cox ring using generators and relations. Both of
these results are in terms of the matroids of all symmetric powers of the
bundle. We also give a criterion for a toric vector bundle to be big and
describe several interesting examples of toric vector bundles which highlights
how positivity properties for toric vector bundles are more complicated than
for toric line bundles.
