2.5. Lines In 3Space And In 2Space
Consider eq(2.1) for the case of 3space.
Writing
P=(
p,q,r
)
,
A=(
a,b,c
)
and
X(
t
)=(
x,y,z
)
,
we have
x=p+ta
y=q+tb
z=r+tc
(2.2)
which are known as scalar parametric equations, or simply, parametric equations, for the line.
For 2space, only the 1^{st} two
equations in (2.2) are required. The
parameter t can then be eliminated to
give
x−p
a
=
y−q
b
Þ
b(
x−p
)−a(
y−q
)=0
(2.3)
which is called a Cartesian equation for the line.
Provided
a≠0
,
we can put it in the pointslope form,
y−q=
b
a
(
x−p
)
where the point
(
p,q
)
and slope
b
a
are easily discerned.
Eq(2.3) can also be written in terms of dot
products, namely,
N⋅(
X−P
)=0
where
N=(
b,−a
)
is called a normal vector to the line since it is perpendicular to the
direction vector A, i.e.,
N⋅A=ba−ab=0
. [cf. Fig.2.2]
Theorem 2.6
Let L
be a line in R^{2} given by
N⋅(
X−P
)=0
and let
d=

P⋅N

‖ N ‖
.
Then
‖ X ‖≤d
∀ X∈L
.
Furthermore,
‖ X ‖ =d
iff X
is the projection of P along N, i.e.,
X=
P⋅N
N⋅N
N
Proof
Since
N⋅X=N⋅P
∀ X∈L
,
we have, by the CauchySchwarz inequality,

N⋅P
 = 
N⋅X

≤

X⋅X

N⋅N

= ‖ X ‖‖ N ‖
Þ
‖ X ‖ ≥

P⋅N

‖ N ‖
=d
Now, the equality holds iff
X=αN
for some
α∈R
. This means
αN⋅N=P⋅N
Þ
α=
P⋅N
N⋅N
Q.E.D.
Corollary
Let Q
be a point not on L, then
‖
X−Q
‖ ≥d
=

(
P−Q
)⋅N

‖ N ‖
∀ X∈L
where the equality applies iff
X−Q
is the projection of
P−Q
and d
is called the distance of Q to L.