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Prelim 2
November 1, 2001
Show all your work.
Your reasoning is important.
No calculators are allowed.
You are allowed
to usea5by8 index card. The number of points each problem is worth is indicated in parantheses. Good
luck.
1. (16 points) Express the vector 6
i
+2
j
as the sum of 2 vectors
a
and
b
such that
a
is parallel to the
vector
i
+
j
and
b
is perpendicular to
i
+
j
.
SOLUTION: Let’s name our vectors;
v
=
i
+
j
and
u
=6
i
j
.Now project
u
onto
v
to get:
proj
v
(
u
)=
u
·
v

v

2
·
v
=
6+2
2
(
i
+
j
)=4
i
+4
j
So now we have
u
=(4
i
j
)+(2
i
−
2
j
) with the vector 4
i
j
parallel to
v
and the vector 2
i
−
2
j
perpendicular to
v
.
2. (16 points) Find the plane that contains the point
P
(1
,
−
2
,
1) and the line:
r
(
t
t
(2
i
j
+
k
i
+
j
+3
k
)
SOLUTION: The normal vector to the plane we want is:
n
=
±
±
±
±
±
±
ijk
241
132
±
±
±
±
±
±
=
i
±
±
±
±
41
32
±
±
±
±
−
j
±
±
±
±
21
12
±
±
±
±
+
k
±
±
±
±
24
13
±
±
±
±
=(8
−
3)
i
−
(4
−
1)
j
+(6
−
4)
k
=5
i
−
3
j
k
So the equation of our plane is:
5(
x
−
1)
−
3(
y
+2)+2(
z
−
1) = 0
3. (16 points) Let
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 Spring '06
 PANTANO

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