## 2.4. Lines And Vector-Valued Functions In *n*-Space

The line
L(
P;A
)
can be considered as the track of a moving
particle with position *X* at time *t* given by

X(
t
)=P+tA
(2.1)

Here
X(
t
)
is an example of a vector-valued function of a
real variable, i.e.,

X:R→L(
P,A
)⊂
R
n
and
t↦X(
t
)

The scalar *t* is often called a **parameter**
and (2.1) a **vector parametric equation**,
or simply, a **vector equation** of the
line
L(
P;A
)
.

Consider a line passing through points *P* and *Q*. By theorem 2.4, it can be
written as
L=L(
P;Q−P
)
so that (2.1) becomes

X(
t
)=P+t(
Q−P
)
=(
1−t
)P+tQ

Note that
X(
a
)=X(
b
)

Û
(
1−a
)P+aQ=(
1−b
)P+bQ

i.e.,
(
a−b
)(
Q−P
)=0

which means
a=b
since
Q−P=A≠0
. Hence, 2 points on the line are distinct iff
their parameters are distinct.

Consider 3 points
X(
a
)
,
X(
b
)
and
X(
c
)
. We say
X(
c
)
is between
X(
a
)
and
X(
b
)
if
a<c<b
or
b<c<a
.

A pair of points *P* and *Q* are **congruent** to another pair *P'* and *Q'* if
‖
P−Q
‖=‖
P'−Q'
‖
. The norm
‖
P−Q
‖
is called the **distance** between *P* and *Q*.