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
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Spring09
Math 231Sec 13.5
Instructor: Yanxiang Zhao
1.2
Symmetric Equations of Lines
If we eliminate the parameter
t
in (
2
) with
a, b, c
are all nonzero, we can get the
symmetric equations
of
L
as follows:
x

x
0
a
=
y

y
0
b
=
z

z
0
c
(3)
If one of
a, b, c
is 0, for instance if
a
= 0, then the 1st equation in (
2
) is unchanged, and
by eliminating
t
from the 2nd and 3rd equation of (
2
), we get the symmetric equations
of
L
as
x
=
x
0
,
y

y
0
b
=
z

z
0
c
(4)
This means that
L
lies in the plane
x
=
x
0
. If
b
= 0, we have
y
=
y
0
,
x

x
0
a
=
z

z
0
c
(5)
If
c
= 0, we have
z
=
z
0
,
x

x
0
a
=
y

y
0
b
(6)
Example
Find the symmetric equations for the line.
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 Spring '08
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 Yanxiang Zhao

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