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.