Scalar Equation of Plane

A little more helpful form of the equations is as follows. Begin with the first form of the vector equation and write a vector for the difference.

{a, b, c} . ({x, y, z}  - {x₀, y₀, z₀})  = 0

{a, b, c} . {x - x₀, y - y₀, z - z₀} = 0

 Now, in fact compute the dot product to obtain,

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

This is known as the scalar equation of plane. Frequently this will be written as,

ax + by + cz = d

in which d = ax₀ + by₀ + cz₀

This second form is frequently how we are given equations of planes. 

Note: If we are given the equation of a plane in this form we can fast get a normal vector for the plane. A normal vector is as follow:

^→n = (a, b, c)