The aim of this paper is to summarise the mathematical basics of the Screw theory and its application in mechanics. The paper provides methodology for finding the degrees of freedom of the effector, as well as for determining its mobility. Furthermore, we give an input-output equation for velocities for the parallel mechanism, that allows to obtain the velocities of the effector and also of the actuated kinematic couples. These algorithms are applied to a planar parallel 3-RTR mechanism where we determine the degrees of freedom and mobility of the moving platform. We determine the input-output equation for velocities, and based on this we investigate the singularities of the moving platform.