UNIT 7
LINES AND PLANES
INTRO: This Unit is about lines and planes: their equations, how they are determined by
points and vectors, and some additional facts about them. In the discussions of this Unit,
x
,
y
and
z
will be variables, and
a
,
b
,
c
and
d
will be constants.
1. Lines
It is very surprising to students that the equation
y
=
mx
+
b
, or equivalently,
ax
+
by
=
c
, is
not necessarily the equation of a line, that is to say, the graph of the equation may not be a
line. It is all a question of dimensions!
Certainly, in two dimensions (in
R
2
) the equation
ax
+
by
=
c
is the equation of a line. We also
say that in the xy plane it is the equation of a line. However, in three dimensions (in
R
3
) it is
the equation of a plane. In fact, in three dimensions any equation of the form
ax
+
by
+
cz
=
d
is the equation of a plane, including, for example, the case when
c
= 0, in which case the
equation becomes
ax
+
by
=
d
, which
in two dimensions
would have been a line, but in three
dimensions is a plane. We will study the equation of a plane in the next section.
What, then, is the equation of a line in three dimensions? It is not possible to describe a line in
three dimensions by a single Cartesian equation, meaning a single equation just in terms of
x
,
y
and
z
(although it is possible to specify a line as the intersection of two Cartesian equations).
For this reason we prefer to give equations of lines in three dimensions as parametric equations.
The equation of a line through the point
P
= (
p
1
, p
2
, p
3
) going in the direction of the vector
~v
=
< v
1
, v
2
, v
3
>
is
x
=
p
1
+
v
1
t
y
=
p
2
+
v
2
t
z
=
p
3
+
v
3
t
We will consider this problem, where the direction
~v
and a single point
P
on the line are given,
as the
model problem
. For such a problem, we can write down the parametric equations of the
line immediately, just as above.
All other problems will be turned into the model problem. Since at least one point on the line
is specified in all problems, the key to any line problem will be TO FIND
~v
!!!
Some examples.
To find the equation of the line through the points
Q
= (
q
1
, q
2
, q
3
) and
R
= (
r
1
, r
2
, r
3
), choose
~v
=
Q

R
VECTOR GEOMETRY
85
GREENBERG