Math 200, Spring 2010
Handout 3
Section 12-2
Parametric Equations of a Line
Section 12-5 Planes in Three-Space
Vector Equation of Line
L
in Three Space (Point-Direction Form)
The line
L
through point
0
0
0
0
(
,
,
)
P x y z
in the direction of
, ,
a b c
=
v
,
where
( )
, ,
P x y z
is an arbitrary point on
L
and
0
r
and
r
are position
vectors of
0
P
and
P
,
is described by
Vector Parametrization:
0
0, 0, 0,
, ,
t
x y z
t a b c
=
+
=
+
r
r
v
The vector
v
is called a
direction vector
for line
L
.
A direction
vector for a line is a substitute for the slope (which only makes
sense in the plane).
However, the direction vector is not unique -
any nonzero scalar multiple of
v
is also a direction vector for
line
L
.
1.
Find a vector equation for the line through the point (3, 1,4)
−
and parallel to the vector 2
7
+ +
i
j
k
.
Parametric and Symmetric Equations of Line
L
in Three Space
The line
L
through point
0
0
0
0
(
,
,
)
P x y z
in the direction of
, ,
a b c
=
v
is described by
Parametric equations:
0
0
0
,
,
x
x
at
y
y
bt
z
z
ct
=
+
=
+
=
+
Symmetric equations:
0
0
0
x
x
y
y
z
z
a
b
c
−
−
−
=
=
The numbers
a
,
b
, and
c
are called direction numbers of
L
.
Any three numbers proportional to
a
,
b
, and
c
could also be used as a set of direction numbers for
L
.
2.