Geometric formulations for the inverse kinematics problem (IKP) in robotics are often set up in the full Cartesian space of three translational and three rotational coordinates (3T3R). When transferring this to tasks with spatial rotation like 3T2R, 2T3R or 2T2R, the result is usually not defined in a minimal set of independent coordinates. Removing the excluded operational space coordinates completely from the expressions is interesting from a theoretical point of view and simplifies further calculations. This can be achieved by formulating a 2R residual using the Z-Y′-X′′ Tait-Bryan angles and a 2T residual derived by the projection of the pointing direction on a plane. In this paper, the minimal-coordinate IKP is derived for 2T2R and 2T3R tasks on position level with application to a gradient-projection scheme. Limitations of the redundant coordinate are considered within the nullspace.