General Planes

A plane in space that is perpendicular to a vector n = < A, B, C > and contains the point P (x₀, y₀, z₀) has the equation

Ax + By + Cz = D

where D = Ax₀ + By₀ + Cz₀.

The Forming of Equation

Consider there is a moving point Q (x, y, z) on the plane with point P (x₀, y₀, z₀). A vector, v can be formed from P to Q and get v = < x-x₀, y-y₀, z-z₀ >. Since n is perpendicular to the plane containing the vector v, n is perpendicular to v as well. As such,

	=	0
	=	0
	=	0
	=	0
	=	0
	=	0
	=	Ax0 + By0 + Cz0
	=	D

Try to move point P below, use a calculator to calculate the value of D at the second line on top of the graph. You will find that the value of D remains unchanged as stated in the first line equation (maybe with a slight difference due to rounding error).

Betty, Created with GeoGebra and Thomas Calculus, Pearson.