sbharris you say the derivation of c force is free have you looked at the derivation it centers around a vector triangle in which the two shorter sides route an actual particle velocity through a point on an imaginary rotating frame of reference the expression xr for the velocity of one of these two components only applies in the limit when it is purely tangential hence the other component which represents the velocity of the particle relative to the rotating frame must be radial when we reach the final expression the coriolis term xv it would indeed be allowed to take on any direction in a plane perpendicular to if v was free but v is not free it is restricted by the derivation to only being a radial velocity