2.17. Linear Cartesian Equations For Planes In R3

Eq(2.16) can be expressed in terms of components.  Setting

        N=( a,b,c )              P=( x 1 , y 1 , z 1 )           X=( x,y,z )

we have

        a( x x 1 )+b( y y 1 )+c( z z 1 )=0                (2.18)

which is known as the Cartesian equation for the plane.  An alternative form is

        ax+by+cz=d                        (2.19)

where

        d=a x 1 +b y 1 +c z 1   =NP

Eq(2.19) is called a linear equation in x, y and z.  In general, a linear equation in R3 represents a plane and vice versa.

 

Two planes with parallel normal vectors are said to be parallel to each other.

Thus, the planes

        ax+by+cz= d 1               ax+by+cz= d 2

are parallel.  Furthermore, the (shortest) distance between them is

  | d 1 d 2 | N = | d 1 d 2 | a 2 + b 2 + c 2

Two planes are orthogonal if their normal vectors are orthogonal.

More generally, the angle between 2 planes is defined as the angle between their normal vectors.