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In one sense there is no instantaneous displacement. There is an instantaneous displacement from theBut we want to prove that the displacement is perpendicular to the force not the velocity.

*origin*, which is the instantaneous position (vector). But, displacement is the difference of two position measurements. As the time increment tends to zero, the displacement over that time increment tends to zero.

The instantaneous quantities must be the time derivative of something. Velocity is the derivative of position and acceleration is the derivative of velocity. Neither of these need tend to zero as the time increment tends to zero.