Say one has two points in the x,y plane. How would one convert those two points lớn a line? Of course I know you could use the slope-point formula và derive the line as following:$$y - y_0 = fracy_1-y_0x_1-x_0(x-x_0)$$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 lớn deduce A, B, C given two points?



Let $P_1:(x_1,y_1)$ & $P_2:(x_2,y_2)$. Then a point $P:(x,y)$ lies on the line connecting $P_1$ và $P_2$ if và 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$$eginvmatrixx_2-x_1 & x-x_1 \y_2-y_1 & y-y_1endvmatrix = 0 Longleftrightarrow(y_1-y_2)x+(x_2-x_1)y+x_1y_2-x_2y_1=0,$$so you get (up lớn scale) $A=y_1-y_2$, $B=x_2-x_1$ & $C=x_1y_2-x_2y_1$.

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