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In an ordered, directional medium, such as liquid crystals or biological membranes, it is often desirable to determine both the degree of ordering (i.e., order parameters) and the directions in space which most aptly characterize it. Here, a method is presented for determining these quantities relative to a laboratory frame. It consists in measuring changes in the light absorbed by a pleochroic dye dissolved in dilute quantity in a directionally ordered medium (e.g., nematic liquid crystal or lipid bilayer) as functions of polarization and angles of incidence. The results of testing this method on a model system of dilute mixtures of dyes in a room temperature nematic liquid crystal are presented. In principle, only six such measurements are necessary to make these determinations; however, some redundancy is desirable to obtain sufficiently accurate results. We show that an accuracy of 5% or better can be obtained from nine measurements. A detailed discussion of the errors that can be expected is given.