An ordered vector basisv_1, ..., v_n for a finite-dimensional vector space V defines an orientation. Another basis w_i = A v_i gives the same orientation if the matrix A has a positive determinant, in which case the basis w_i is called oriented. Any vector space has two possible orientations since the determinant of an nonsingular matrix is either positive or negative. For example, in R^2, {e_1, e_2} is one orientation and {e_2, e_1}~{e_1, - e_2} is the other orientation. In three dimensions, the cross product uses the right-hand rule by convention, reflecting the use of the canonical orientation {e_1, e_2, e_3} as e_1×e_2 = e_3.

bundle orientation | manifold orientation