The idea of a velocity vector comes from classical physics. By representing the position and motion of a single particle using vectors, the equations for motion are simpler
and more intuitive. Suppose the position of a particle at time 
is given by the position vector 
. Then the velocity vector 
is the derivative of the position,
 |
For example, suppose a particle is confined to the plane and its position is given by 
.
Then it travels along the unit circle at constant speed. Its velocity vector is 
. In a diagram, it makes
sense to translate the velocity vector so it originates
at 
.
In particular, it is drawn as an arrow from 
to 
.
Another example is a particle traveling along a hyperbola specified parametrically by 
. Its velocity vector is then given by
, illustrated above.
Travel down the same path, but using a different function is called a reparameterization, and the chain rule describes the change in velocity.
For example, the hyperbola can also be parametrized
by 
.
Note that 
,
and by the chain rule, 
.
Note that the set of possible velocity vectors forms a vector space. If 
and 
are two paths through the origin, then so is 
and the velocity vector of this path is 
. Similarly, if 
is a scalar, then the path 
has velocity vector 
. It makes sense to distinguish the velocity vectors at
different points. In physics, the set of all velocity vectors gives all possible
combinations of position and momentum, and is called phase space. In mathematics,
the velocity vectors form the tangent space, and the collection of tangent spaces
forms the tangent bundle.
