MAC2313 Exam 1 Version A Solution
1. Find a unit vector orthogonal to the vectors
~a
=
h
3
,
1
,

1
i
and
~
b
=
h
1
,
0
,
3
i
. (10 points)
Solution:
There are two unit vectors that are orthogonal to
~a
and
~
b
. Let
~v
=
~a
×
~
b
=
h
3
,

8
,
1
i
.
~v
is not a unit vector since

~v

=
√
74. The 2 unit vectors orthogonal to
~a
and
~
b
are
~v

~v

=
3
√
74
,

8
√
74
,
1
√
74

~v

~v

=

3
√
74
,
8
√
74
,

1
√
74
.
2. Let
~a,
~
b
be vectors in
R
3
and let
orth
~a
(
~
b
) =
~
b

proj
~a
(
~
b
) be the orthogonal projection of
~
b
onto
~a
. Prove that
orth
~a
(
~
b
) is orthogonal to
~a
. (15 points)
Solution:
To show that two vectors are orthogonal, we must show that the dot product
is 0. Remember that
~a
•
~a
=

~a

2
,
~a
•
orth
~a
(
~
b
) =
~a
•
(
~
b

proj
~a
(
~
b
)) =
~a
•
~
b

~a
•
proj
~a
(
~
b
) =
~a
•
~
b

~a
•
~a
•
~
b

~a

2
!
~a
=
~a
•
~
b

~a
•
~
b

~a

2
!
~a
•
~a
=
~a
•
~
b


~a

2

~a

2
!
(
~a
•
~
b
) = 0
.
3. Find the equation and sketch the graph of 2 surfaces in which the curve
~
r
(
t
) =
cos(
t
)
,
sec(
t
)
,
sin
2
(
t
)
lies. (Sketch the surfaces on two separate graphs). (15 points)
Solution:
Three surfaces can be quickly identified using relations between two component
functions of
~
r
.
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 Spring '08
 Keeran
 Calculus, Linear Algebra, Geometry, Vectors, Dot Product

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