## 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 *R*^{n}^{}

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,t∈R
}

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
X∈L(
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(
Q−P
)+t(
R−P
)
}
(2.4)

#### Proof

### Theorem 2.11

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

#### Proof