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} - {x_{0}, y_{0}, z_{0}}) = 0
{a, b, c} . {x - x_{0}, y - y_{0}, z - z_{0}} = 0
Now, in fact compute the dot product to obtain,
a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0
This is known as the scalar equation of plane. Frequently this will be written as,
ax + by + cz = d
in which d = ax_{0} + by_{0} + cz_{0}
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)