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Mathematics 192, Fall 2006: Solutions to Prelim Exam 1
1. (a) The vectors
v
=
i
+ 2
j
+
a
k
and
w
=
i
+
j
+
k
are perpendicular if and only if
v
·
w
= 0 = 3 +
a
, so
a
=

3.
(b) The area of the parallelogram spanned by
v
and
w
is equal to

v
×
w

=

(2

a
)
i
+ (
a

1)
j

k

=
√
2
a
2

6
a
+ 6
This is equal to
√
6 if
a
= 0 or
a
= 3.
2. (a) Two vectors parallel to
P
are
~
BA
=
h
1
,
1
,
1
i
and
~
CA
=
h
0
,
0
,
2
i
. A normal
vector is
~
BA
×
~
CA
=
h
2
,

2
,
0
i
. A unit normal vector to the plane is therefore
1
√
2
h
1
,

1
,
0
i
. Another correct answer is:

1
√
2
h
1
,

1
,
0
i
.
(b) The equation of
P
is
x

y
= 1. A set of parametric equations for the line are
x
= 1 +
t, y
= 1

t, z
= 1. We substitute these into the equation for
P
and
solve for
t
: (1 +
t
)

(1

t
) = 1, yielding
t
= 1
/
2. The point of intersection is
(3
/
2
,
1
/
2
,
1).
3. (a) Compute
v
(
t
) =
t
j
+
t
2
k
, so
s
(
t
) =
Z
t
0

v
(
u
)

du
=
Z
t
0
u
√
1 +
u
2
du
=
1
3
(1 +
t
2
)
3
2

1
3
.
The substitution
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 Fall '06
 PANTANO
 Multivariable Calculus, Vectors

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