Mathematics 23 Midterm Exam, September 30, 2004 + Solutions
Answer all questions and be sure to
show all work
. No notes, books, or calculators
are allowed.
1. Let
a
=
i
+
j
+
2k
and let
b
=

3i

2j
+
4k
. Are
a
and
b
orthogonal, parallel,
or neither? Explain carefully.
Answer. Since
a
·
b
=

1
, the vectors are not orthogonal. They are not parallel,
since
a
is not a scalar multiple of
b
. So they are skew.
2. a. Find the unit tangent vector
T
(
t
) to the curve
r
(
t
) =
h
6
t
5
,
4
t
3
,
2
t
i
.
Answer. a.
r
0
(
t
) =
h
30
t
4
,
12
t
2
,
2
i
and

r
0
(
t
)

=
√
900
t
8
+ 144
t
4
+ 4. So
T
(
t
) =
(
√
900
t
8
+ 144
t
4
+ 4)

1
h
30
t
4
,
12
t
2
,
2
i
.
b. Find the parametric equations of the tangent line to
r
(
t
) at the point (6
,
4
,
2).
Answer.
t
= 1 at the point (6
,
4
,
2). The direction numbers for the line are
r
0
(1) =
h
30
,
12
,
2
i
.
So the parametric equations become
x
(
t
) = 6+30
t,y
(
t
) = 4+12
t,
and
z
(
t
) = 2 + 2
t
.
3. a. Find an equation of the plane
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 Spring '06
 YUKICH
 Math, Equations, Vector Space, Parametric Equations, Acceleration, Parametric equation

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