Say one has two points in the x,y plane. How would one convert those two points to a line? Of course I know you could use the slope-point formula & derive the line as following: $$y - y_0 = \frac{y_1-y_0}{x_1-x_0}(x-x_0)$$ However this manner obviously doesn't hold when $x_1-x_0 = 0$ (vertical line). The more generic approach should however be capable of define every line (vertical line would simply mean B = 0); $$Ax+By +C = 0$$
But how to deduce A, B, C given two points?
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$\begingroup$Let $P_1:(x_1,y_1)$ and $P_2:(x_2,y_2)$. Then a point $P:(x,y)$ lies on the line connecting $P_1$ and $P_2$ if and only if the area of the parallellogram with sides $P_1P_2$ and $P_1P$ is zero. This can be expressed using the determinant as$$ \begin{vmatrix} x_2-x_1 & x-x_1 \\ y_2-y_1 & y-y_1 \end{vmatrix} = 0 \Longleftrightarrow (y_1-y_2)x+(x_2-x_1)y+x_1y_2-x_2y_1=0, $$so you get (up to scale) $A=y_1-y_2$, $B=x_2-x_1$ and $C=x_1y_2-x_2y_1$.
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