We present a systematic derivation of the extended Jeffery's orbit for rigid ellipsoidal and V-shaped polymer molecules in linear incompressible viscous flows using a Lagrange multiplier's method based on a constraining force argument. It reproduces the well-known Jeffery's orbit for rotating ellipsoids. The method is simple and applicable to any rigid body immersed in a linear flow field so long as a discrete set of representative points on the rigid body can be identified that possess the same rotational degrees of freedom as the rigid body itself. The kinematics of a single V-shaped rigid polymer driven by a linear flow field are discussed, where steady states exist along with time-periodic states in limited varieties. Finally, we show how the kinematics of the rigid V-shaped polymer can be used in the derivation a kinetic theory for the solution of rigid biaxial liquid crystal polymers, where Brownian motion, excluded volume interaction and flow driven kinematics are included.