2. Lines

We usually represent lines in parametric form: x$= P _ 0 + t$   v, where $P _ 0$ is a fixed point, v is a fixed vector giving the direction, and $t$ is a parameter, which we can regard as time.

This is the same as saying (in three dimensions) $$\left \{ \begin{array}{c} x = p + a t\\ y = q + b t\\ z = r + c t \end{array}\right .$$
where $P _ 0 = (p,q,r)$ and v$= (a,b,c)$.