Logic behind normal line in expressing plane

Question

why do we consider normal line in expressing a plane,say in $R^3$, of the form $ ax + by + cz = d $? What is the logic behind this normal line selection? Plz provide intuitive explanations.Thanks

Answer 1

For me the most intuitive explanation has been as follows, say you have a point on a plain T0 (x0, y0, z0). Then a vector between any point (x, y, z) on the plane and T0 and the normal vector (a,b, c) should have their scalar product equal to 0. Then

(x-x0)a + (y-y0)b + (z-z0)c=0,

Defining d= ax0+by0+cz0, completes the explanation