2.16. Normal Vectors To Planes In R3

Definition: Normal Vector

A non-zero vector N R 3  is a normal vector to the plane M={ P+sA+tB }  if N is perpendicular to both A and B.

Theorem 2.15

Consider a plane M={ P+sA+tB }  in R3.  Let N=A×B .  Then

(a)     N is a normal vector to M.

(b)     M={ X|  ( XP )N=0 }

Proof

(a)  Follows from properties (d) and (e) of Theorem 2.12.

(b)   Proof is analogous to that for the line discussed in section 2.5.

Theorem 2.16

Given a plane M through P with a normal vector N and

        d= | PN | N                         (2.17)

Then, X d   XM , where the equal sign applies iff

        X=tN      with       t= PN NN

Proof

Proof is analogous to that for the line discussed in Theorem 2.6.

Corollary

The shortest distance from a point Q to a plane M is

        d= | ( QP )N | N

Actually, d is called the distance from Q to M.