লম্বত্বের মাধ্যমে একটি রেখার সমীকরণ

One approach to defining lines algebraically is via "slopes" (basically directions), but that doesn't extend well to higher dimensions.

The better approach is actually in terms of orthogonality.

Parametrically, a line takes the form $\{(a+tu,b+tv)\}$ over reals $t$. The line goes in the nonzero direction $\pm(u,v)$.

So the $(tu,tv)$ component is orthogonal to the direction $(-v,u)$. Thus our line is alternatively described as the set of $(x,y)$ such that $-v(x-a)+ u(y-b) = 0$: the set of $(x,y)$ such that $(x-a,y-b)$ is orthogonal to $(-v,u)$ (because being orthogonal to $(-v,u)$ is equivalent to having direction $\pm(u,v)$).

Cleaning up constants, let $A = -v$, $B = u$, and $C = va-ub$. Then the line here is simply the set of $(x,y)$ satisfying $Ax+By+C = 0$.

Conversely, for any constants $A,B,C$ with $A,B$ not both zero (why does this matter?), it's easy to see that $Ax+By+C = 0$ defines a line. (Why?)