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Student Number:
Math 20C.
Midterm Exam 1
October 17, 2005
Read each question carefully, and answer each question completely.
Show all of your work. No credit will be given for unsupported answers.
Write your solutions clearly and legibly. No credit will be given for illegible solutions.
1. (6 points) Consider the vectors
v
=
h
6
,
2
,

3
i
and
w
=
h
2
,
2
,
1
i
.
(a) Find a vector normal to both,
v
and
w
.
v
×
w
¯
¯
¯
¯
¯
¯
i
j
k
6
2

3

2
2
1
¯
¯
¯
¯
¯
¯
= (2 + 6)
i

(6

6)
j
+ (12 + 4)
k
=
h
8
,
0
,
16
i
.
A solution is any vector proportional to
v
×
w
. For example
u
=
h
1
,
0
,
2
i
.
(b) Find the area of the parallelogram formed by
v
and
w
.
A
=

v
×
w

=
p
8
2
+ 16
2
=
p
8
2
(1 + 4) = 8
√
5
.
(c) Find a vector of length one in the direction of
w
.

w

=
√
4 + 4 + 1 =
√
9 = 3
.
u
=
w

w

=
1
3
h
2
,
2
,
1
i
.
#
Score
1
2
3
4
Σ
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View Full Document2. (6 points) Find an equation for the plane that passes through the points (

1
,
1
,
1),
(

1
,

1
,
1), and (0
,
0
,
2).
Let
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 Fall '08
 Helton
 Math, Calculus, #, Euclidean geometry

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