Parametric equation of a straight line ($r=a+st,s\in \mathbb{R}$)

A straight line through a point $a$ in nonzero direction $t$ is the set of vectors $r=a+st$ for real parameter $s$. If $|t|=1$, changes in the parameter equal signed distances along the line. Two such lines intersect exactly when their two parameter formulas give a common vector for some choice of their parameters.