If 
is the vector joining the origin 
of a coordinate system and the point 
, then specification of the vector function 
defines 
, 
and 
as function of 
. As changes, the terminal point of describes a space curve having parametric equations 
, 
, 
. If the parameter is the arc length 
measured from some fixed point on the curve, then
is a unit vector in the direction of the tangent to the curve and is called the unit tangent vector. If is the time 
, then
is the velocity with which the terminal point of describes the curve. We have
from which we see that the magnitude of 
, often called the speed, is 
. Similarly,
is the acceleration with which the terminal point of describes the curve. These concepts have important applications in mechanics.
