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We study the orientability of vector bundles with respect to a family of cohomology theories called EO-theories. The EO-theories are higher height analogues of real K-theory KO. For each EO-theory, we prove that the direct sum of i copies of any vector bundle is EOorientable for some specific integer i. Using a splitting principal, we reduce to the case of the canonical line bundle over ℂ ℙ∞. Our method involves understanding the action of an order p subgroup of the Morava stabilizer group on the Morava E-theory of ℂ ℙ∞. Our calculations have another application:We determine the homotopy type of the S¹-Tate spectrum associated to the trivial action of S¹ on all EO-theories. [ABSTRACT FROM AUTHOR]