2.7. Planes In Euclidean n-Space

Since 2 lines define a plane, the linear span L( A,B )  is a plane defined by the lines L( 0,A )  and L( 0,B ) .  To make the plane pass through a point P, one needs only to add the vector P to every vector in L( A,B ) .

Definition: Plane in Rn

A set M of points is called a plane if there exists a point P and 2 L.I. vectors A and B such that

        M={ P+sA+tB|s,tR }

More specifically, M is called a plane through P spanned by A and B.

Theorem 2.7

Two planes M={ P+sA+tB }  and M'={ P+sC+tD }  through the same point P are equal iff L( A,B )=L( C,D ) .

Proof

But the definition of a plane, it is obvious that L( A,B )=L( C,D )  Þ M=M' .

Conversely, if M=M' , then for any given s and t, there exist s' and t' such that

        P+sA+tB=P+s'C+t'D            Þ     sA+tB=s'C+t'D

i.e., L( A,B )=L( C,D ) .  Q.E.D.

Theorem 2.8

Two planes M={ P+sA+tB }  and M'={ Q+sA+tB }  spanned by the same vectors A and B are equal iff Q is on M.

Proof

 

Definition: Parallelism

Two planes M={ P+sA+tB }  and M'={ P+sC+tD }  are said to be parallel if L( A,B )=L( C,D ) .  Also, a vector X is parallel to the plane M if XL( A,B ) .

Theorem 2.9

Given a point Q not on plane M, there is one and only one plane M' that contains Q and parallel to M.

Proof

 

Theorem 2.10

Let P, Q, R be 3 points not on the same line.  There is one and only one plane M containing these 3 points.  Furthermore,

M={ P+s( QP )+t( RP ) }                       (2.4)

Proof

 

Theorem 2.11

3 vectors A, B, C are L.I. iff they lie on the same plane through O.

Proof