Álgebra Linear I - Poli - P1 - 2012

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Questão Alternativa 
1 B 
2 C 
3 B 
4 A 
5 C 
6 E 
7 C 
8 A 
 9 D 
10 A 
11 B 
12 A 
13 E 
14 A 
15 B 
16 A 
Q
1
.
S
ej
a
fi
x
ad
a
u
m
a
or
ie
n
ta
c¸a˜
o
d
e
V
3
e
se
ja
E
u
m
a
b
a
se
or
to
n
o
rm
a
l
p
o
si
ti
va
d
e
V
3
.
C
on
si
d
er
e
os
ve
to
re
s
~a
=
(1
,2
,1
) E
,
~ b
=
(1
,1
,0
) E
,
~c
=
(0
,1
,0
) E
e
~ d
=
(1
,0
,1
) E
.
S
ej
a
~w
=
α
~a
+
β
~ b
+
γ
~c,
co
m
α
,β
,γ
∈
R
.
S
e
p
ro
j ~ d
~w
=
−
~ d
e
~w
∧
(~c
+
~ d
)
=
(0
,2
,−
2
) E
,
en
ta˜
o
p
o
d
e-
se
afi
rm
ar
q
u
e
α
+
β
−
γ
e´
ig
u
al
a
(a
)
0.
(b
)
−4
.
(c
)
−2
.
(d
)
2.
(e
)
4
.
Q
2
.
S
ej
am
o
s
ve
to
re
s
u
n
it
a´r
io
s
~u
,~v
∈
V
3
ta
is
q
u
e
a
m
ed
id
a
d
o
aˆ
n
g
u
lo
en
tr
e
~u
e
~v
e´
pi
/3
.
S
e
λ
∈
R
,
p
o
d
em
os
afi
rm
ar
q
u
e
‖λ
~u
+
~v
‖=
√ 3
se
,
e
so
m
en
te
se
,
(a
)
λ
=
2
ou
λ
=
−2
.
(b
)
λ
=
1
ou
λ
=
−1
.
(c
)
λ
=
1
ou
λ
=
−2
.
(d
)
λ
=
2
ou
λ
=
−1
.
(e
)
λ
=
3
ou
λ
=
−3
.
Q
3
.
C
on
si
d
er
e
o
tr
iaˆ
n
gu
lo
A
B
C
il
u
st
ra
d
o
n
a
fi
gu
ra
ab
ai
x
o
:
�
�
�
�
�
�
�
��
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HH
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
XX
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A
B
C
X
D
r
r
O
p
on
to
X
es
ta´
so
b
re
o
se
gm
en
to
A
C
e
o
p
o
n
to
D
es
ta´
so
b
re
o
se
g
m
en
to
B
X
.
S
e
X
e´
o
p
on
to
m
e´d
io
d
o
se
gm
en
to
A
C
,
−−→ BD
=
2 3
−−→ BX
e
3
−−→ AD
=
α
−−→ AB
+
β
−−→ BC
,
co
m
α
,β
∈
R
,
p
o
d
e-
se
afi
rm
ar
q
u
e
α
+
β
e´
ig
u
al
a
(a
)
4.
(b
)
3.
(c
)
5
.
(d
)
2.
(e
)
1
.
Q
4
.
S
ej
a
fi
x
a
d
a
u
m
a
or
ie
n
ta
c¸a˜
o
em
V
3
.
S
ej
a
{~u
,~v
}
u
m
su
b
co
n
ju
n
to
li
n
e-
a
rm
en
te
in
d
ep
en
d
en
te
d
e
V
3
fo
rm
ad
o
p
o
r
ve
to
re
s
u
n
it
a´
ri
o
s.
C
o
n
si
d
er
e
a
s
se
gu
in
te
s
afi
rm
ac¸
o˜e
s:
(I
)
(~u
·~v
)2
+
‖~u
∧~
v
‖2
=
1.
(I
I)
A
a´r
ea
d
o
p
ar
al
el
og
ra
m
o
d
et
er
m
in
ad
o
p
o
r
~u
e
~u
∧~
v
e´
√ 1
−
(~u
·~v
)2
.
(I
II
)
~u
∧~
v
e´
u
m
a
co
m
b
in
ac¸
a˜o
li
n
ea
r
d
e
~u
e
~v
.
P
o
d
e-
se
afi
rm
ar
q
u
e
(a
)
ap
en
a
s
as
afi
rm
a
c¸o˜
es
(I
)
e
(I
I)
sa˜
o
se
m
p
re
ve
rd
a
d
ei
ra
s.
(b
)
ap
en
as
as
afi
rm
a
c¸o˜
es
(I
)
e
(I
II
)
sa˜
o
se
m
p
re
v
er
d
ad
ei
ra
s.
(c
)
ap
en
as
a
afi
rm
a
c¸a˜
o
(I
)
e´
se
m
p
re
ve
rd
ad
ei
ra
.
(d
)
ap
en
as
as
afi
rm
a
c¸o˜
es
(I
I)
e
(I
II
)
sa˜
o
se
m
p
re
ve
rd
ad
ei
ra
s.
(e
)
ap
en
as
a
afi
rm
a
c¸a˜
o
(I
I)
e´
se
m
p
re
ve
rd
ad
ei
ra
.
Q
5
.
S
ej
a
{~v
1
,~v
2
,~v
3
}u
m
a
b
a
se
d
e
V
3
.
S
ej
a
~v
4
∈
V
3
ta
l
q
u
e
~v
2
=
3~v
1
+
3~
v 3
−~v
4
.
S
e
λ
∈
R
,
en
ta˜
o
{λ
~v
1
+
~v
2
−~v
3
,~v
3
+
~v
4
,~v
3
−~v
4
}e´
u
m
a
b
as
e
d
e
V
3
se
,
e
so
m
en
te
se
,
(a
)
λ
=
−3
.
(b
)
λ
6=
3.
(c
)
λ
6=
−3
.
(d
)
λ
6=
0
.
(e
)
λ
=
3
.
Q
6
.
S
ej
a
E
u
m
a
b
as
e
or
to
n
or
m
al
d
e
V
3
.
C
o
n
si
d
er
e
u
m
tr
iaˆ
n
gu
lo
A
B
C
ta
l
q
u
e
−−→ AB
=
(1
,−
1
,1
) E
e
−→ AC
=
(α
,α
,α
) E
,
co
m
α
∈
R
,α
6=
0.
P
o
d
e-
se
a
fi
rm
a
r
q
u
e
a
m
ed
id
a
d
a
al
tu
ra
d
es
se
tr
iaˆ
n
gu
lo
re
la
ti
va
a`
b
a
se
A
B
e´
ig
u
a
l
a
2
√
3
3
se
,
e
so
m
en
te
se
,
(a
)
α
=
√ 2
ou
α
=
−√
2.
(b
)
α
=
√ 3
ou
α
=
−√
3.
(c
)
α
=
1
ou
α
=
−1
.
(d
)
α
=
√
3
2
ou
α
=
−
√
3
2
.
(e
)
α
=
√
2
2
ou
α
=
−
√
2
2
.
Q
7
.
C
on
si
d
er
e
o
cu
b
o
A
B
C
D
E
F
G
H
il
u
st
ra
d
o
n
a
fi
g
u
ra
a
b
a
ix
o
:
�
�
�
�
�
�
�
�
�
�
�
�
A
BF
E
D
CG
H
S
ej
am
E
e
F
as
b
as
es
d
e
V
3
d
a
d
as
p
or
E
=
{ 2
−−→ AB
,−−
→
B
H
,−
−→ AC
} ,
F
=
{ −
→
A
C
,2
−→ AG
,−
2
−−→ AH
} .
S
e
~v
∈
V
3
e
~v
=
(1
,2
,3
) F
,
en
ta˜
o
a
so
m
a
d
as
co
o
rd
en
ad
a
s
d
e
~v
n
a
b
a
se
E
e´
ig
u
al
a
(a
)
−1
.
(b
)
2.
(c
)
−2
.
(d
)
1.
(e
)
6.
Q
8
.
C
on
si
d
er
e
o
cu
b
o
A
B
C
D
E
F
G
H
d
e
ar
es
ta
u
n
it
a´
ri
a
il
u
st
ra
d
o
n
a
fi
g
u
ra
ab
ai
x
o:
�
�
�
�
�
�
�
�
�
�
�
�
A
BF
E
C
DH
G
M
S
ej
a
M
o
p
on
to
m
e´d
io
d
o
se
gm
en
to
B
D
e
co
n
si
d
er
e
o
es
p
ac¸
o
V
3
or
ie
n
ta
d
o
d
e
m
o
d
o
q
u
e
a
b
as
e
{ −
−→ AB
,−
→
A
E
,−
−→ AM
}
se
ja
p
os
it
iv
a.
S
ej
a
~w
=
3−−
→
A
B
+
5−
→
A
E
+
−−→ AM
.
A
ss
in
al
e
a
a
lt
er
n
at
iv
a
q
u
e
co
n
te´
m
o
va
lo
r
d
o
p
ro
d
u
to
es
ca
la
r
( −
→
A
F
∧−
−→ AH
) ·
~w
.
(S
u
g
e
st
a˜
o
:
F
ac¸
a
ca´
lc
u
lo
s
em
te
rm
os
d
e
co
or
d
en
a
d
as
n
a
b
a
se
{ −
−→ AB
,−
→
A
E
,−
→
A
C
} .
)
(a
)
−1
.
(b
)
0.
(c
)
2.
(d
)
−2
.
(e
)
1.
Q
9
.
S
ej
a
fi
x
ad
a
u
m
a
or
ie
n
ta
c¸a˜
o
d
e
V
3
e
se
ja
E
=
{~e
1
,~e
2
,~e
3
}
u
m
a
b
a
se
or
to
n
or
m
al
p
os
it
iv
a
d
e
V
3
.
C
on
si
d
er
e
os
ve
to
re
s
~u
=
(1
,1
,−
1)
E,
~v
=
(1
,0
,1
) E
,
~w
=
(1
,6
,1
) E
.
E
n
ta˜
o
,
a
so
m
a
d
as
co
o
rd
en
ad
as
d
e
~w
n
a
b
a
se
F
=
{~u
,~v
,~u
∧~
v
}e´
ig
u
al
a
(a
)
−5
.
(b
)
5.
(c
)
3.
(d
)
1.
(e
)
−1
.
Q
1
0
.
Sej
a
E
u
m
a
b
a
se
o
rt
on
or
m
al
d
e
V
3
e
co
n
si
d
er
e
o
s
ve
to
re
s
~u
=
(1
,−
1
,1
) E
e
~v
=
(1
,0
,1
) E
.
S
ej
a
m
a
,b
,c
∈
R
.
S
e
o
ve
to
r
~w
=
(a
,b
,c
) E
e´
ta
l
q
u
e
‖~w
‖=
3
,
~w
e´
o
rt
o
g
on
a
l
a
~v
e
o
co
ss
en
o
d
a
m
ed
id
a
d
o
aˆn
gu
lo
en
tr
e
~u
e
~w
e´
1/
3
,
en
ta˜
o
p
o
d
e
a
fi
rm
a
r
q
u
e
(a
)
a
=
√ 3
o
u
a
=
−√
3.
(b
)
a
=
3
ou
a
=
−3
.
(c
)
a
=
3√
3
o
u
a
=
−3
√ 3
.
(d
)
a
=
√
3
3
ou
a
=
−
√
3
3
.
(e
)
a
=
1/
3
ou
a
=
−1
/3
.
Q
1
1
.
S
ej
a
fi
x
ad
a
u
m
a
or
ie
n
ta
c¸a˜
o
em
V
3
.
S
ej
a
m
~u
,~v
,
~w
∈
V
3
ve
to
re
s
n
a˜
o
-
n
u
lo
s,
d
oi
s
a
d
oi
s
d
is
ti
n
to
s
en
tr
e
si
.
P
o
d
e-
se
a
fi
rm
a
r
q
u
e
(a
)
{~u
,~v
,
~w
}
e´
li
n
ea
rm
en
te
in
d
ep
en
d
en
te
se
,
e
so
m
en
te
se
,
q
u
a
is
q
u
er
d
oi
s
ve
to
re
s
d
e
{~u
,~v
,
~w
}s
a˜o
or
to
go
n
ai
s
en
tr
e
si
.
(b
)
~u
+
~v
e
~u
−
~v
sa˜
o
o
rt
og
on
ai
s
se
,
e
so
m
en
te
se
,
‖~u
‖=
‖~v
‖.
(c
)
~u
+
~v
e
~u
−
~v
sa˜
o
p
a
ra
le
lo
s
se
,
e
so
m
en
te
se
,
~u
e
~v
sa˜
o
or
to
g
on
a
is
.
(d
)
{~u
,~v
,
~w
}e´
li
n
ea
rm
en
te
in
d
ep
en
d
en
te
se
,
e
so
m
en
te
se
,
~w
=
~u
∧~
v
.
(e
)
{~u
,~v
,
~w
}e´
li
n
ea
rm
en
te
d
ep
en
d
en
te
se
,
e
so
m
en
te
se
,
{~u
,~v
}e´
li
n
ea
rm
en
te
d
ep
en
d
en
te
.
Q
1
2
.
S
ej
a
E
u
m
a
b
a
se
o
rt
on
or
m
al
d
e
V
3
e
co
n
si
d
er
e
o
s
ve
to
re
s:
~a
=
(1
,1
,1
) E
,
~ b
=
(1
,2
,0
) E
,
~c
=
(4
,1
,−
1)
E,
~v
=
(x
,y
,z
) E
,
co
m
x
,y
,z
∈
R
.
S
e
‖~v
‖=
2√
11
,
~v
e´
or
to
g
o
n
a
l
a
~c
e
{~a
,~ b
,~v
}
e´
li
n
ea
rm
en
te
d
ep
en
d
en
te
,
p
o
d
e-
se
afi
rm
ar
q
u
e
|x
+
y
+
z
|e´
ig
u
a
l
a
(a
)
6.
(b
)
2.
(c
)
3
.
(d
)
4.
(e
)
5
.
Q
1
3
.
S
ej
a
fi
x
a
d
a
u
m
a
or
ie
n
ta
c¸a˜
o
em
V
3
.
S
ej
am
~u
,~v
,
~w
∈
V
3
ve
to
re
s
n
a˜
o
-
n
u
lo
s.
C
on
si
d
er
e
as
se
gu
in
te
s
afi
rm
a
c¸o˜
es
:
(I
)
p
ro
j ~w
~u
=
p
ro
j ~w
~v
se
,
e
so
m
en
te
se
,
~u
=
~v
.
(I
I)
{ ~u
,~v
,~u
∧
(~u
∧~
v
)}
e´
li
n
ea
rm
en
te
d
ep
en
d
en
te
.
(I
II
)
{~u
+
~v
,~u
−
~v
,~u
∧
~v
}
e´
u
m
a
b
as
e
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