Several exact and numerical techniques have been established to solve inverse (electromagnetic) scattering problems connected with cylindrical bodies whose orientations are known in advance. In these cases, by choosing the coordinate system properly, one can reduce the problem to a two-dimensional scalar problem. On the other hand, when the orientations are not known, it is unavoidable that the problem will be three-dimensional and vector-valued and, as far as we know, such problems have not been well studied. The aim of this paper is to solve the inverse (electromagnetic) scattering problems connected with bodies buried in a half-space with unknown orientations, and in particular, to determine the orientation of the body. Once the orientation is obtained. other properties can be recovered using known methods. The method given in the present work is based on the Born approximation, which can always be ensured by choosing the frequency of the incident wave appropriately.