Math Preliminaries (continued)

Vector and Parametric Equations vs. Cartesian Equations 

Recall from statics that we used a position vector R  to locate a point P in space and wrote it, in cartesian coordinates, as follows:

If point P moves along a path, however, we can track its location by expressing the position vector  R(t), and thus, its x and y components,  as function(s) of the parameter time.  Expressing  R(t) in cartesian vector form, we get the vector equation of a curve   where,   x = f(t)  and  y = g(t)  are called the parametric equations of the curve.

 A   cartesian path equation  y = f(x)   may be obtained by eliminating time t from  the x = f(t) and y  = g(t) parametric equations. 

While it is more customary to give the equation of a curve in cartesian path form, it alone is not a description of motion—it is only a locus of (x,y) values. 

Parametric equations, on the other hand,  provide a complete description of a body’s motion, giving information such as: 
        (a)  Starting point, 
        (b)  Direction, 
        (c)  Rate (velocity), 
        (d)  Domain (the part of the path the particle moves along), and 
        (e)  Acceleration.