A bundle orientation of a real vector bundle 
is a compatible choice of vector space orientation in its fibers.
It can be specified by a covering by trivializations 
such
that the transition functions are vector
space orientation-preserving. Over a paracompact Hausdorff base 
, this is equivalent to the top exterior
power 
having a nowhere-zero bundle section, which specifies
an orientation by choosing a positive ray in each fiber.
Real projectivization identifies both rays and
therefore does not specify an orientation. A bundle is
called orientable if there exists an orientation. Under this hypothesis on the base space, a bundle 
of bundle rank 
is orientable iff 
is a trivial line
bundle (Hatcher 2017, p. 25).
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.
