Logic behind normal line in expressing plane

It is a convenient way to express the position and orientation of the plane with a single equation.
There is no ambiguity in the selection apart from an overall scaling of the equation (you can multiply a,b,c,d by a non-zero constant without changing the plane).

 

The right hand rule might be a suitable illustration why it is useful.

 

For a plane in ##\mathbb{R}^3## whose equation is ax + by + cz = d, the vector ##\vec{N} = <a, b, c>## is perpendicular to (or normal to) the plane. The normal gives the orientation of the plane in the three-dimensional space, but that same vector is perpendicular to an infinite family of parallel planes. If you also know a point on the plane, that narrows the choices down to a single plane.

 

It is easy to use it in calculations.

 

Think about it in terms of the geometry of the situtation. If you have a vector in ##\mathbb{R}^3##, it is perpendicular to an infinite number of planes, all with the same orientation (and therefore parallel). A given value of d identifies one and only one of these planes.

If d is unknown, then yes, we have a family of parallel planes.

For the algebra, let's say we know a vector N = <a, b, c> that is perpendicular to the plane, and a point ##P_0(x_0, y_0, z_0)## that is on the plane. Position the vector so that its tail is at P₀, as in the drawing below.
Take an arbitrary point on the plane P(x, y, z) that is different from P₀ and form the vector ##\vec{P_0P} = <x - x_0, y - y_0, z - z_0>##. Since ##\vec{P_0P}## and ##\vec{N}## are perpendicular, their dot product must be zero. IOW, ##\vec{P_0P} \cdot \vec{N} = 0##, so ##(x - x_0) \cdot a + (y - y_0) \cdot b + (z - z_0) \cdot c = 0##. The equation ax + by + cz = d comes directly from this dot product.

 

I'm not sure which vectors you're talking about -- the normal or a vector in the plane.

The question as you originally asked it didn't have anything to do with graphing a plane. If you have the equation of a plane, it's easy to find three points in the plane, and you can use these points to sketch the plane.

Or, you can find the intersection of the plane in the three coordinate planes.