When a particle moves in the space, during a time-interval I, we think of its coordinates as functions of the variable t (= time), namely (x=x(t),y=y(t),z=z(t)). The set of all the locations of the particle is a curve in the space, named the trajectory of the particule, and the functions x,y,z define a parametrization of .
The vector is the position vector of the particle at time t, and the point M such that is the position of the particle at time t.
Without the z-component, we have the corresponding definitions for a curve/ the motion of a particule in the plane.
We have: , i.e. the trajectory of the particle is on the parabola whose equation is y=1-2x2. Is the trajectory the whole parabola? No! because . With the notations of Figure 2 the trajectory is the arc AC, on which the particule travels alternatively in both directions, with a period of .