In this paper we introduce a new distance for robustly matching vectors of 3D rotations. A special representation of 3D rotations, which we coin full-angle quaternion (FAQ), allows us to express this distance as Euclidean. We apply the distance to the problems of 3D shape recognition from point clouds and 2D object tracking in color video. For the former, we introduce a hashing scheme for scale and translation which outperforms the previous state-of-the-art approach on a public dataset. For the latter, we incorporate online subspace learning with the proposed FAQ representation to highlight the benefits of the new representation.