## 2.2. Lines In *n*-Space

*R*^{n} is an analytic model of *n*-D
Euclidean space *E*^{n}. Thus, the geometric concepts and properties
of *E*^{n} are expressed in
terms of *n*-tuples of real numbers in *R*^{n}.

### Definition (Point)

A **point**
is a vector (*n*-tuple) in *R*^{n}.

### Definition (Line)

Let *P*
be a point and *A* a non-zero vector.

A **line**
through *P* and parallel to *A* is the set of points

L(
P;A
)={
P+tA:t∈R
}
={
P+tA
}

The vector *A* is called the **direction
vector** of the line.

A point *Q*
is **on** the line
L(
P;A
)
if
Q=P+tA
for some *t*.

### Properties

The line
L(
O;A
)
is the linear span
L(
A
)
of *A*.

Thus, we can write
L(
P;A
)={
P+X:X∈L(
A
)
}
. [See Fig.2.1]