S
ECTION
1.4
L
INES AND
P
LANES
, P
ART
I
9
1.4.
L
INES AND
P
LANES
, P
ART
I
In two dimensions, there are numerous ways of presenting the equation of a line, such as
point-slope form
,
y
y
0
m
x
x
0
. In this form we are specifying a point on the line,
x
0
, y
0
, and the slope of the line,
m
.
In three dimensions, one way to specify a line is with a point,
x
0
, y
0
, z
0
, on the line,
and the
direction
of the line, which we specify with a vector
m
m
1
, m
2
, m
3
.
L
m
P
0
x
0
, y
0
, z
0
P
x, y, z
You should think of an equation for such a line as a “point-testing procedure”. If the
equation holds for a point
x, y, z
, then
x, y, z
lies on the line, and otherwise
x, y, z
does
not lie on the line. From this viewpoint, the point
P
x, y, z
lies on the line
L
if and only
if the vector
P
0
P
is parallel to the direction vector
m
. Remembering that two vectors are
parallel if and only if they are scalar multiples, we see that
P
lies on the line if and only if
P
0
P
m
t
for some scalar
t
. Expanding
P
0
P
, we see that
x, y, z
lies on the line if and only if
x
x
0
, y
y
0
, z
z
0
m
t,
or
x, y, z
m
t
x
0
, y
0
, z
0
.
This is called a
vector-valued function
, or simply a
vector function
, because it takes one num-
ber as input,
t
, and returns a vector,
x, y, z
.
You should think of a vector function as
tracing out a curve consisting of all the points its vectors point to from the origin.
Vector Equation for a Line.
The
vector
equation
for
the
line
through the point
x
0
, y
0
, z
0
parallel to the vector
m
is
r
t
m
t
x
0
, y
0
, z
0
,
where
t
.
By solving the vector equation for a line for
x
,
y
, and
z
, we obtain another form.