In this chapter, we will consider motion in two dimensions taken to be the X-Y plane, for convenience.
The figure shows a particle at time t moving along a curved path. Its position or displacement from the origin is measured by
the vector
,
its velocity is indicated by the vector
which
must be tangential to the path of the particle. The acceleration is indicated by the
vector which does not bear any unique relationship to the
path of the particle but depends on the rate at which the velocity
changes
with time as the particle moves along its path.
The vectors are interrelated and can be
expressed in terms of their components using unit vectors as shown below:
These equations can easily be extended to three dimensions by adding
to them, the terms
respectively, in which
is the unit vector along the Z-direction.
,
its velocity is indicated by the vector
which
must be tangential to the path of the particle. The acceleration is indicated by the
vector 
which does not bear any unique relationship to the
path of the particle but depends on the rate at which the velocity
are interrelated and can be
expressed in terms of their components using unit vectors as shown below:
respectively, in which
is the unit vector along the Z-direction.