## 2.3. Some Simple Properties Of Straight Lines In *R*^{n}

### Definition (Parallelism of Vectors)

Two vectors *A* and *B* are parallel iff
A=αB
for some
α∈R
.

### Theorem 2.1

Two lines
L(
P;A
)
and
L(
P;B
)
are equal iff *A* and *B* are parallel.

#### Proof of Þ

Let
L(
P;A
)=L(
P;B
)
. A point *X*
on the line can therefore be written as

X=P+αA=P+βB
for some
α,β∈R

Hence,
A=γB
where
γ=
β
α
∈R
. That is, *A*
and *B* are parallel.

#### Proof of Ü

Let *A*
and *B* be parallel so that
A=αB
for some
α∈R
.

If *Q*
is on
L(
P;A
)
,
then
Q=P+tA
for some
t∈R
. Thus,

Q=P+tA=P+tαB=P+βB
where
β=tα∈R

Hence, *Q*
is also on
L(
P;B
)
. Since this is true for arbitrary *Q*, we have
L(
P;A
)⫅L(
P;B
)
. Reversing the role of *A* and *B* gives
L(
P;B
)⫅L(
P;A
)
. Hence,
L(
P;A
)=L(
P;B
)
.

### Definition (Parallelism of Lines)

Two lines
L(
P;A
)
and
L(
P;B
)
are parallel if *A* and *B* are parallel.

### Theorem 2.3 (Euclid's Parallel Postulate)

Given a line *L* and a point *Q* not on *L*, there is one and only one line *L'* through *Q* and parallel to *L*.

#### Proof

Let *A
*be the direction vector of *L*. We can write
L'=L(
Q,A
)
. According to theorem 2.1, *L'* is unique. Q.E.D.

### Theorem 2.4 (2 distinct points determines a lines)

If
P≠Q
,
there is 1 and only 1 line containing both *P*
and *Q*.

The line can be written as
{
P+t(
Q−P
)
}
.

#### Proof

The line
L(
P;Q−P
)={
P+t(
Q−P
)
}
goes through both *P* and *Q* since they
corresponds to
t=0
and
t=1
,
respectively.

Let *L'*
be another line that goes through both *P*
and *Q*. Since *L'*
goes through *P*, we have
L'=L(
P;A
)
for some nonzero *A*. Now, *L'* also goes through *Q* so
that
P+αA=Q
for some *α*.
Hence,
αA=Q−P
so that *A*
and *Q**-P* are parallel. By theorem 2.2,
L'=L(
P;Q−P
)
. Q.E.D.

### Example

From theoem 2.4, we see that *Q* is on
L(
P,A
)
iff *A*
and *Q**-P* are parallel, i.e.,
Q−P=αA
.

Consider the 3-space case with
P=(
1,2,3
)
and
A=(
2,−1,5
)
.

To test if
Q=(
1,1,4
)
is on
L(
P,A
)
,
we examine
Q−P=(
0,−1,1
)
and find that it is not a multiple of *A*.
Hence, *Q* is not on the line.

For
Q=(
5,0,13
)
,
we have
Q−P=(
4,−2,10
)=2A
so that *Q*
is on *L*.

### Theorem 2.5 (Linear Dependence)

Two vectors
A,B∈
R
n
are linearly dependent iff they lie on the
same line through the origin.

#### Proof

If either vector is zero, the theorem holds
trivially.

If both are nonzero, they are dependent iff
B=αA
for some
α∈R
.

However,
B=αA
iff *B*
is on the line
L(
0,A
)
. Q.E.D.