MATH 20C Lecture 5  Monday, October 4, 2010
Planes
1. The plane through 3 points,
P
1
,P
2
,P
3
.
A point
P
= (
x,y,z
) is in the plane if and only if the volume of the parallelipiped with sides
→
P
1
P,
→
P
1
P
2
,
→
P
1
P
3
has volume 0 (drew picture). This is the same as saying that
det
→
P
1
P
→
P
1
P
2
→
P
1
P
3
= 0
Example
Take
P
1
= (0
,
1
,
0)
,P
2
= (1
,
1
,
0)
,P
3
= (1
,
0
,
0)
.
The plane through these 3 points
has equation
det
h
x,y

1
,z
i
h
1
,
0
,
0
i
h
1
,

1
,
0
i
= 0
which is to say
z
= 0
.
This is the
xy
plane.
Note
: In general the equation of a plane in space has the form
ax
+
by
+
cz
=
d
2. The plane through the origin perpendicular to
→
N
=
h
1
,
5
,
10
i
Drew a picture. A point
P
= (
x,y,z
) is in this plane if and only if
→
OP
⊥
→
N,
which is to say
→
OP
·
→
N
= 0
.
This means
h
x,y,z
i · h
1
,
5
,
10
i
= 0
,
which gives
x
+ 5
y
+ 10
z
= 0
.
3. The plane through
P
0
= (2
,
1
,

1) and perpendicular to
→
N
=
h
1
,
5
,
10
i
Drew a picture. A point
P
= (
x,y,z
) is in this plane if and only if
→
P
0
P
⊥
→
N,
which is to say
→
P
0
P
·
→
N
= 0
.
This means
h
x

2
,y

1
,z
+ 1
i · h
1
,
5
,
10
i
= 0
,
which gives
x
+ 5
y
+ 10
z
=

3
.
This plane is parallel to the plane in the previous example. In both cases, the coeﬃcients of
x,y,z
are the components of the vector
→
N.
In the case of
x
+ 5
y
+ 10
z
=

3 one gets the constant

3 by plugging in the coordinates of
the point
P
0
in the left hand side.
Deﬁnition
A vector perpendicular to a plane
P
is called a normal vector to that plane. Note
that this is implies that all normal vectors to a given plane are proportional.
So the coeﬃcients of
x,y,z
are the components of a normal vector to the plane. Conversely,
if the equation of the plane is
ax
+
by
+
cz
=
d,
then
h
a,b,c,
i
is a normal vector to it.
1