Plane versus 3D coordinates We know that line $ax+by+c=0$ is one dimensional and the plane $ax+by+cz+d=0$ is two dimensional.
My question is if line is one dimensional so why 2D points $(x, y)$ are used for line? And if plane two dimensional so why 3D points $(x, y, z)$ are used for plane?
N. B. - I want to understand the intuition rather than details proof.

As noted in the comment of Bernard,  we use two coordinates when we work on a plane (2D) and three coordinates when we work on the space (3D). So the equation $ax+by+d=0$ represents a stright line if it is refferred to points on a plane , but, if it is referred to points in a 3D space, it is a special case of $ax+by+cz=d$, with $c=0$ and it represents a plane parallel to the $z$ axis .
Also, in 3D the (two) equations
$$
\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}
$$
represent a straight line.
