Computationally efficient analyses are presented which provide rapid decisions making capabilities for determining whether various types of planned motions may be successfully executed. Algorithms are based upon a geometrical quantification of the manipulator dexterity and workspace, and as far as the authors are aware these algorithms are new. They enable the determination of possible motions during the task planning stage without requiring the computation of individual joint angles which produce the desired end-effector trajectory. This paper investigates essentially rectilinear motion using by way of example, a planar 3R manipulator which provides a proper analytical foundation for extension to various types of spatial manipulators.