How can we derive the standard form of the linear equation: $Ax+By+C$? How can we derive the standard form of the linear equation: $Ax+By+C=0$? What do "$A$", "$B$" and "$C$" in the standard form of the linear equation mean? As in the point-slope form of the linear equation: $y-y_0=m(x-x_0)$, $m$ is the slope, and $x$, $y$ are the coordinates of any point. Likewise, $x_0$, $y_0$ are the coordinates of the given point. 

If you interpret it as a vector equation, it makes a lot of sense...
$$
\begin{pmatrix}A&B\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \mathbf{V}\cdot\mathbf{r} = -C
$$
where $\mathbf{r}$ is the position vector and $\mathbf{V}$ is a vector containing the two coefficients. This can be interpreted as "the set of points $\mathbf{r}$ whose length in the direction of $\mathbf{V}$ is $\frac{-C}{|\mathbf{V}|}$". That is, $\mathbf{V}$ is a vector perpendicular to the line. If you divide through by $\sqrt{A^2+B^2}$, you find that $|\mathbf{V}|=1$ and $\mathbf{V}$ is the unit vector normal to the line, while $-C$ is the distance from the origin to the point on the line closest to the origin.
This works in 3D, as well, where you now have $\mathbf{V}$ being the unit vector normal to the plane.
