2
MATH 23, FALL 2011 REVIEW SHEET SOLUTIONS
Solution:
The standard way to do this is to nd a vector parallel to both planes, which
is the same as being perpendicular to both normals. So, the crossproduct of the two
normals will work. That is, the vector
A
=
N
1
×
N
2
= (
i
+
j
)
×
(
j

k
)
=
k
+
j

i
=

i
+
j
+
k
will be in the direction of the line of intersection of both planes. In order to nd the
equation of this line, you need a point on the line; a point on both planes. But, if
y
= 0
,
then
x
= 2
and
z
=

4
, so
(2
,
0
,

4)
is on the line. Then, the equation for the line is
r
(
t
) = 2
i

4
k
+
t
(

i
+
j
+
k
)
.
An alternate way to solve this problem is to note that you can set
y
=
t
in the equation
for both planes, and solve for
x
in the rst plane, and
z
in the second, giving
x
=
2

t, y
=
t,
and
z
=
t

4
.
(5) Find an equation of the plane
P
through the points
(2
,
1
,
0)
,
(1
,
1
,
1)
, and
(0
,
3
,

1)
.
Solution: