LESSON 3
Planes and Quadric Surfaces
READ:
Sections 13.5, 13.6
NOTES:
Equations of planes in 3space play an important role in vector calculus. Recall that one way to specify a line
in the
x
,
y
plane is to name a point on the line and a slope for the line. From those two items, an equation
for the line can be determined. A plane in 3space can also be speciﬁed by giving two pieces of information
about the plane. One is a vector in a direction perpendicular
to the plane, say
→
n
=
h
a,b,c
i
, and the second
item is a single point,
P
0
= (
x
0
,y
0
,z
0
), on the plane. If you picture things in your mind, you will see that
when the vector
→
n
has been speciﬁed, there will be a stack of planes to which that vector is perpendicular,
sort of like sheets of paper sandwiched together in a pad. The given point that the plane must go through
selects one particular sheet from the pad. The vector
→
n
is called a
normal vector
for the plane. Notice
that the magnitude of
→
n
is not important; its only job is to determine a direction. If the normal vector is
multiplied by any nonzero scalar, the new vector serves just as well as a normal vector for the plane. Often a
normal vector is divided by its length to produce a
unit normal vector
since that makes many formulas look
a little neater. Notice that there are just two unit normals to a plane, and they are negatives of each other.
The particular point speciﬁed on the plane also does not matter. Any point on the plane can be used.
The description of a plane via a normal vector and a point on the plane can be translated into a variety of
equations that are satisﬁed by points
P
= (
x,y,x
) on the plane and no other points.
Vector form:
Using the notation above, let’s use
P
0
for the point (
x
0
,y
0
,z
0
), and
P
for the point (
x,y,z
).
Since the point
P
0
is on the plane, we see that
P
is on the plane if and only if the line segment from
P
0
to
P
is in the plane, and that’s correct if and only if
→
n
is perpendicular to that line segment, and that’s so if and
only if
→
n
is perpendicular to a vector in the direction of that line segment. Since
→
P
0
P
=
h
x

x
0
,y

y
0
,z

z
0
i
is a vector in the direction of that line segment, it follows that the point
P
= (
x,y,z
) is on the plane if and
only if
→
n
·
→
P
0
P
= 0, since that dot product tests to see if the two vectors are perpendicular. In other words,
P
is on the plane if and only if
→
n
·
→
P
0
P
= 0
or
→
n
·
→
P
=
→
n
·
→
P
0
.
Either of those equations is called a