 Solution

Set $\Delta x=x_B-x_A$ and $\Delta y=y_B-y_A$. A displacement from $A$ lies along the straight line precisely when its two-dimensional determinant with $(\Delta x,\Delta y)$ vanishes. Thus an implicit equation isEquivalently, write $ax+by+c=0$, with $a=\Delta y$, $b=-\Delta x$ and $c=\Delta x{y}_A-\Delta y{x}_A$. The vector $(a,b)$ is a normal vector. The formula handles vertical and horizontal straight lines without division. It presupposes $A\ne B$: coincident points do not determine a unique straight line.