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NAME:
Spring 2011, MAC 2313, Quiz 1
UFID:
Section: 3124
1. Find a unit vector that is orthogonal to both
i
+
j
and
i
+
k
. (2 points)
Solution.
Let
v
be the cross product of
i
+
j
and
i
+
k
, then we know
v
is
perpendicular to both of them. Now
v
= (
i
+
j
)
×
(
i
+
k
) =
±
±
±
±
±
±
i j k
1 1 0
0 1 1
±
±
±
±
±
±
=
i

j
+
k
,
(1)
and

v

=
p
1
2
+ (

1)
2
+ 1
2
=
√
3. Hence
v

v

=
1
√
3
i

1
√
3
j
+
1
√
3
k
(2)
is a unit vector that is perpendicular to both
i
+
j
and
i
+
k
.
2. Find the volume of the parallelepiped determined by the vectors
a
=
i
+
j
+
k
,
b
=
i

j
+
k
, and
c
=

i
+
j
+
k
. (2 points)
Solution.
Since
a
·
(
b
×
c
) =
±
±
±
±
±
±
1
1
1
1

1 1

1
1
1
±
±
±
±
±
±
=

4
,
(3)
we know that the volume is

a
·
(
b
×
c
)

= 4.
3. Determine whether the lines
L
1
and
L
2
are parallel, skew, or intersecting. If they
intersect, ﬁnd the point of intersection. Here
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This note was uploaded on 12/15/2011 for the course MAC 2313 taught by Professor Keeran during the Spring '08 term at University of Florida.
 Spring '08
 Keeran
 Calculus, Geometry

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