VETORES - 2

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Faculdade Esta´cio do Recife
CA´LCULO VETORIAL E
GEOMETRIA ANALI´TICA
Prof. Se´rgio Barreto
L I S T A D E E X E R C I´ C I O S - V E T O R E S 2
1. Encontre o produto interno
−→
u • −→v nos casos abaixo:
(a)
−→
u =
(
2, 1
)
e
−→
v =
(
−4, 2
)
(b)
−→
u =
(
−1, 4
)
e
−→
v =
(√
3,
√
2
)
(c)
−→
u =
(
−2, 0
)
e
−→
v =
(
3,−1
)
(d)
−→
u =
(
4, 3
)
e
−→
v =
(
4, 5
)
(e)
−→
u =
(
4, 7, 3
)
e
−→
v =
(
2, 1, 1
)
(f)
−→
u =
(
0,−5, 2
)
e
−→
v =
(
3, 2,−6
)
(g)
−→
u =
(
7, 1,−7
)
e
−→
v =
(
0, 1,−1
)
(h)
−→
u =
(
−2,−1, 3
)
e
−→
v =
(
−4, 3,−1
)
2. Encontre a norma do vetor
−→
v nos casos abaixo:
(a)
−→
v =
(
−4, 1
)
(b)
−→
v =
(√
3,
√
2
)
(c)
−→
v =
(
3, 0
)
(d)
−→
v =
(
4, 4
)
(e)
−→
v =
(
7, 2, 1
)
(f)
−→
v =
(
3,−5,−6
)
(g)
−→
v =
(
7, 7,−7
)
(h)
−→
v =
(
−4, 3,−1
)
3. Ate´ que ponto deve-se prolongar o segmento de extremos A = (2, 0, 0) e B = (0, 2, 0) , no
sentido de A para B, de modo que seu comprimento quadruplique?
4. Dado
−→
v =
(
a,−2
)
, calcular os valores de a para que se tenha ‖−→v ‖= 3.
1
5. Verifique quais dos vetores abaixo e´ unita´rio.
(a)
−→
v =
(
1
2
,
1
2
)
(b)
−→
v =
(
1
3
,
2
3
)
(c)
−→
v =
(√
3
2
,
1
2
)
(d)
−→
v =
(
3
5
,−4
5
)
(e)
−→
v =
(√
3
2
,
1
2
,
1
2
)
(f)
−→
v =
(√
3
2
,
1
2
, 0
)
(g)
−→
v =
(√
5
4
,
1
4
,
√
10
4
)
(h)
−→
v =
(
1
3
,
2
3
,−1
)
6. Encontre os versores dos vetores abaixo:
(a)
−→
v =
(
−4, 1
)
(b)
−→
v =
(√
3,
√
2
)
(c)
−→
v =
(
3, 0
)
(d)
−→
v =
(
4, 4
)
(e)
−→
v =
(
7, 2, 1
)
(f)
−→
v =
(
3,−5,−6
)
(g)
−→
v =
(
7, 7,−7
)
(h)
−→
v =
(
−4, 3,−1
)
7. Encontre o aˆngulo entre os vetores
−→
u e
−→
u nos casos:
(a)
−→
u =
(
2, 1
)
e
−→
v =
(
4, 6
)
(b)
−→
u =
(
7, 5
)
e
−→
v =
(
1, 2
)
(c)
−→
u =
(
−2, 0
)
e
−→
v =
(
3,−1
)
(d)
−→
u =
(
4, 3
)
e
−→
v =
(
4, 5
)
(e)
−→
u =
(
2, 1, 4
)
e
−→
v =
(
4, 6,−5
)
(f)
−→
u =
(
−3, 1,−2
)
e
−→
v =
(
0, 1,−1
)
(g)
−→
u =
(
1, 1,−3
)
e
−→
v =
(
1,−1, 2
)
(h)
−→
u =
(
−2,−1, 3
)
e
−→
v =
(
−4, 3,−1
)
2
8. Encontre
−→
u × −→v nos casos abaixo:
(a)
−→
u =
(
2,−1, 3
)
e
−→
v =
(
4, 1, 1
)
(b)
−→
u =
(
0, 2, 0
)
e
−→
v =
(
1, 3,−1
)
(c)
−→
u =
(
1, 1, 2
)
e
−→
v =
(
3, 1,−1
)
(d)
−→
u =
(
2, 1, 0
)
e
−→
v =
(
0, 1,−1
)
(e)
−→
u =
(
2, 1, 4
)
e
−→
v =
(
4, 6,−5
)
(f)
−→
u =
(
−3, 1,−2
)
e
−→
v =
(
0, 1,−1
)
(g)
−→
u =
(
1, 1,−3
)
e
−→
v =
(
1,−1, 2
)
(h)
−→
u =
(
−2,−1, 3
)
e
−→
v =
(
−4, 3,−1
)
9. Calcular, se poss´ıvel, as a´reas do paralelogramo e do triaˆngulo determinados pelos vetores
−→
u
e
−→
v , nos casos abaixo:
(a)
−→
u =
(
2,−1, 3
)
e
−→
v =
(
4, 1, 1
)
(b)
−→
u =
(
0, 2, 0
)
e
−→
v =
(
1, 3,−1
)
(c)
−→
u =
(
1, 1, 2
)
e
−→
v =
(
3, 1,−1
)
(d)
−→
u =
(
2, 1, 4
)
e
−→
v =
(
4, 6,−5
)
(e)
−→
u =
(
−3, 1,−2
)
e
−→
v =
(
0, 1,−1
)
(f)
−→
u =
(
1, 1,−3
)
e
−→
v =
(
1,−1, 2
)
10. Encontre
(−→
u ,
−→
v ,
−→
w
)
, produto misto dos vetores
−→
u ,
−→
v e
−→
w , nos casos abaixo:
(a)
−→
u =
(
0, 2, 1
)
,
−→
v =
(
1, 3, 4
)
e
−→
w =
(−1, 4, 2).
(b)
−→
u =
(
1, 1, 3
)
,
−→
v =
(
2,−1, 5) e −→w = (4,−3, 1).
(c)
−→
u =
(
2, 4,−5), −→v = (7,−1, 0) e −→w = (2, 2, 3).
11. Verificar se A, B, C e D sa˜o pontos coplanares nos casos abaixo:
(a) A =
(
1, 1, 0
)
, B =
(
0, 2, 3
)
, C =
(
2, 0,−1) e D = (−1, 3, 5).
(b) A =
(
2, 3, 4
)
, B =
(
1,−1, 9), C = (5,−3, 7) e D = (0, 3, 6).
(c) A =
(
1, 1, 1
)
, B =
(
1, 2, 1
)
, C =
(
3, 0, 1
)
e D =
(
5, 7, 10
)
.
12. Calcular, se poss´ıvel, os volumes do paralelep´ıpedo e do tetraedro determinados pelos vetores
−→
u ,
−→
v e
−→
w , nos casos abaixo:
(a)
−→
u =
(
0, 2, 1
)
,
−→
v =
(
1, 3, 4
)
e
−→
w =
(−1, 4, 2).
(b)
−→
u =
(
1, 1, 3
)
,
−→
v =
(
2,−1, 5) e −→w = (4,−3, 1).
(c)
−→
u =
(
2, 4,−5), −→v = (7,−1, 0) e −→w = (2, 2, 3).
3