Spring09
Math 231Sec 13.5
Instructor: Yanxiang Zhao
Section 13.5: Equations of Lines and Planes
1 Equation of Lines
1.1 Parametric Equations of Lines
A line
L
can be determined by the direction of
L
and a typical point on it. Mathe
matically, consider a line
L
in 3d space whose direction is parallel to
v
, and a point
P
0
(
x
0
,y
0
,z
0
) sitting on
L
(Figure 01)
. How to describe the line
L
analytically by using
v
and
P
0
(
x
0
,y
0
,z
0
) ?
Lemma 1.1
Two vectors
a
and
b
are parallel if and only if there exists a constant
t
such that
a
=
t
b
. Symbolically,
a
k
b
⇔
a
=
t
b
,
where
t
is determined by
t
=

b


a

.
Let
v
be a vector parallel to
L
. Let
P
(
x,y,z
) be an arbitrary point on
L
and let
r
0
and
r
be the position vectors of
P
0
and
P
(namely, they have representations
→
OP
0
and
→
OP
). Then the geometric vector
→
P
0
P
corresponds the algebraic vector
r

r
0
.
Obviously,
r

r
0
is parallel to
v
, applying
Lemma 1.1
yields
r

r
0
=
t
v
or equivalently
r
=
r
0
+
t
v
(1)
Equation (1) is called
vector equation
of
L
, where
t
is the parameter. Each value of
t
gives the position vector
r
of a point on
L
. In other words, as
t
varies, the line is
traced out by the tip of the vector
r
.
If
v
= (
a,b,c
), and
r
0
= (
x
0
,y
0
,z
0
)
,
r
= (
x,y,z
), then we can rewrite (1) componen
twisely as follows:
x
=
x
0
+
at,
y
=
y
0
+
bt,
z
=
z
0
+
ct
(2)
and equation (2) is called
parametric equations
of
L
.
Example
Find the vector equation and parametric equations for the line.
•
The line through the point (6
,

5
,
2) and parallel to the vector
h
1
,
3
,

2
/
3
i
.
•
The line through the points (1
,
3
,
2) and (

4
,
3
,
0).
1