A real vector bundle 
has an orientation if there exists a covering by
trivializations 
such that the transition
functions are vector space orientation-preserving.
Alternatively, there exists a section of the projectivization
of the top exterior power of the bundle, 
. A bundle is called orientable if there exists an
orientation. Hence a bundle 
of bundle rank 
is orientable iff 
is a trivial line
bundle.
An orientation of the tangent bundle is equivalent to an orientation on the base manifold. Not all bundles are orientable, as can be seen by the tangent bundle of the Möbius strip. The nontrivial line bundle on the circle is also not orientable.
