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Math 20C.
Final Exam
June 15, 2006
No calculators or any other devices are allowed on this exam.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
Read each question carefully. If any question is not clear, ask for clariFcation.
Answer each question completely, and show all your work.
1. (10 points) Find the plane through the point
P
0
= (2
,
1
,

1) which is perpendicular to
the planes 2
x
+
y

z
= 3 and
x
+ 2
y
+
z
= 2.
The plane is determined by its normal vector
n
and a point.
We choose the point to be
P
0
= (2
,
1
,

1). The normal vector can be computed as
n
=
n
1
×
n
2
,
n
1
=
h
2
,
1
,

1
i
,
n
1
=
h
1
,
2
,
1
i
.
where
n
1
and
n
2
are the normal vectors to the planes 2
x
+
y

z
= 3 and
x
+ 2
y
+
z
= 2,
respectively. Then,
n
=
¯
¯
¯
¯
¯
¯
i
j
k
2
1

1
1
2
1
¯
¯
¯
¯
¯
¯
=
h
(1 + 2)
,

(2 + 1)
,
(4

1)
i
⇒
n
=
h
3
,

3
,
3
i
.
We can pick up any vector proportional to
h
3
,

3
,
3
i
as the normal vector to the plane, for
example a simpler one is
n
=
h
1
,

1
,
1
i
. Then, the equation of the plane is
(
x

2)

(
y

1) + (
z
+ 1) = 0
⇒
x

y
+
z
= 0
.