Dados os vetores ~u = (1,−1), ~v = (−3, 4) e ~w = (8,−6), calcular:
(a) |~u|
(b) |~v|
(c) |~w|
(d) |~u + ~v|
(e) |2~u − ~w|
(f) |~w − 3~u|
(g)
~v
|~v|
(h)
∣∣∣∣∣ ~u|~u|
∣∣∣∣∣
a
b
c
d
e
f
g
h
|~u|
|~v|
|~w|
|~u + ~v|
|2~u − ~w|
|~w − 3~u|
~v/|~v|
|~u|~u|

Question

Dados os vetores ~u = (1,−1), ~v = (−3, 4) e ~w = (8,−6), calcular:
(a) |~u|
(b) |~v|
(c) |~w|
(d) |~u + ~v|
(e) |2~u − ~w|
(f) |~w − 3~u|
(g)
~v
|...

Ed · Answer

Para calcular as magnitudes (normas) dos vetores e as operações solicitadas, vamos passo a passo:

(a) Para calcular |~u|:
$$
|~u| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
$$

(b) Para calcular |~v|:
$$
|~v| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
$$

(c) Para calcular |~w|:
$$
|~w| = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
$$

(d) Para calcular |~u + ~v|:
$$
~u + ~v = (1, -1) + (-3, 4) = (1 - 3, -1 + 4) = (-2, 3)
$$
$$
|~u + ~v| = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
$$

(e) Para calcular |2~u − ~w|:
$$
2~u = 2(1, -1) = (2, -2)
$$
$$
2~u - ~w = (2, -2) - (8, -6) = (2 - 8, -2 + 6) = (-6, 4)
$$
$$
|2~u - ~w| = \sqrt{(-6)^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
$$

(f) Para calcular |~w − 3~u|:
$$
3~u = 3(1, -1) = (3, -3)
$$
$$
~w - 3~u = (8, -6) - (3, -3) = (8 - 3, -6 + 3) = (5, -3)
$$
$$
|~w - 3~u| = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
$$

(g) Para calcular $\frac{~v}{|~v|}$:
$$
\frac{~v}{|~v|} = \frac{(-3, 4)}{5} = \left(-\frac{3}{5}, \frac{4}{5}\right)
$$

Agora, temos os resultados:
- (a) |~u| = $\sqrt{2}$
- (b) |~v| = 5
- (c) |~w| = 10
- (d) |~u + ~v| = $\sqrt{13}$
- (e) |2~u − ~w| = $2\sqrt{13}$
- (f) |~w − 3~u| = $\sqrt{34}$
- (g) $\frac{~v}{|~v|} = \left(-\frac{3}{5}, \frac{4}{5}\right)$

Se precisar de mais alguma coisa, é só avisar!

Ed · Answer

(a) |~u| = √(1² + (-1)²) = √2
(b) |~v| = √((-3)² + 4²) = 5
(c) |~w| = √(8² + (-6)²) = 10
(d) |~u + ~v| = |(1-3, -1+4)| = √((-2)² + 3²) = √13
(e) |2~u − ~w| = |(2-8, -2+6)| = √((-6)² + 4²) = 2√13
(f) |~w − 3~u| = |(8-3, -6+3)| = √(5² + (-3)²) = √34
(g) ~v/|~v| = (-3/5, 4/5)
(h) |~u|~u| = √2(1,-1) = (√2, -√2)

Geometria Analítica

Lista_2_GAAL_2019

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Lista_2_GAAL_2019

Dados os vetores ~u = 2~i − 3~j, ~v =~i − ~j e ~w = −2~i + ~j, determinar:
(a) 2~u − ~v
(b) ~v − ~u + 2~w
(c)
1
2
~u − 2~v − ~w
(d) 3~u − 1
2
~v − 1
2
~w
a
b
c
d
2~u − ~v
~v − ~u + 2~w
1/2~u − 2~v − ~w
3~u − 1/2~v − 1/2~w

Dados os vetores ~u = (3,−1), ~v = (−1, 2), determinar o vetor ~x tal que:
(a) 4(~u − ~v) + 1/3~x = 2~u − ~x
(b) 3~x − (2~v − ~u) = 2(4~x − 3~u)
a
b
4(~u − ~v) + 1/3~x = 2~u − ~x
3~x − (2~v − ~u) = 2(4~x − 3~u)

Dados os pontos A(−1, 3), B(2, 5), C(3,−1) e O(0, 0), calcular:
(a) −−→OA − −→AB
(b) −→OC − −→BC
(c) 3−→BA − 4−→CB
a
b
c
−−→OA − −→AB
−→OC − −→BC
3−→BA − 4−→CB

Dados os vetores ~u = (2,−4), ~v = (−5, 1) e ~w = (−12, 6), determinar a1 e a2 tais que
~w = a1 · ~u + a2 · ~v

~w = a1 · ~u + a2 · ~v

Dados os pontos A(3,−4) e B(−1, 1) e o vetor ~v = (−2, 3), calcular:
(a) (B − A) + 2~v
(b) (A − B) − ~v
(c) B + 2(B − A)
(d) 3~v + 2(A − B)
a
b
c
d
(B − A) + 2~v
(A − B) − ~v
B + 2(B − A)
3~v + 2(A − B)

Encontrar o vértice oposto a B, no paralelogramo ABCD, para:
(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)
a
b
Encontrar o vértice oposto a B, no paralelogramo ABCD, para:
(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)

Mais conteúdos dessa disciplina

Geometria Analítica

Lista_2_GAAL_2019

Respostas

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Lista_2_GAAL_2019

Dados os vetores ~u = 2~i − 3~j, ~v =~i − ~j e ~w = −2~i + ~j, determinar:(a) 2~u − ~v(b) ~v − ~u + 2~w(c)12~u − 2~v − ~w(d) 3~u − 12~v − 12~wabcd2~u − ~v~v − ~u + 2~w1/2~u − 2~v − ~w3~u − 1/2~v − 1/2~w

Dados os vetores ~u = (3,−1), ~v = (−1, 2), determinar o vetor ~x tal que:(a) 4(~u − ~v) + 1/3~x = 2~u − ~x(b) 3~x − (2~v − ~u) = 2(4~x − 3~u)ab4(~u − ~v) + 1/3~x = 2~u − ~x3~x − (2~v − ~u) = 2(4~x − 3~u)

Dados os pontos A(−1, 3), B(2, 5), C(3,−1) e O(0, 0), calcular:(a) −−→OA − −→AB(b) −→OC − −→BC(c) 3−→BA − 4−→CBabc−−→OA − −→AB−→OC − −→BC3−→BA − 4−→CB

Dados os vetores ~u = (2,−4), ~v = (−5, 1) e ~w = (−12, 6), determinar a1 e a2 tais que~w = a1 · ~u + a2 · ~v~w = a1 · ~u + a2 · ~v

Dados os pontos A(3,−4) e B(−1, 1) e o vetor ~v = (−2, 3), calcular:(a) (B − A) + 2~v(b) (A − B) − ~v(c) B + 2(B − A)(d) 3~v + 2(A − B)abcd(B − A) + 2~v(A − B) − ~vB + 2(B − A)3~v + 2(A − B)

Encontrar o vértice oposto a B, no paralelogramo ABCD, para:(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)abEncontrar o vértice oposto a B, no paralelogramo ABCD, para:(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)

Mais conteúdos dessa disciplina

Dados os vetores ~u = 2~i − 3~j, ~v =~i − ~j e ~w = −2~i + ~j, determinar:
(a) 2~u − ~v
(b) ~v − ~u + 2~w
(c)
1
2
~u − 2~v − ~w
(d) 3~u − 1
2
~v − 1
2
~w
a
b
c
d
2~u − ~v
~v − ~u + 2~w
1/2~u − 2~v − ~w
3~u − 1/2~v − 1/2~w

Dados os vetores ~u = (3,−1), ~v = (−1, 2), determinar o vetor ~x tal que:
(a) 4(~u − ~v) + 1/3~x = 2~u − ~x
(b) 3~x − (2~v − ~u) = 2(4~x − 3~u)
a
b
4(~u − ~v) + 1/3~x = 2~u − ~x
3~x − (2~v − ~u) = 2(4~x − 3~u)

Dados os pontos A(−1, 3), B(2, 5), C(3,−1) e O(0, 0), calcular:
(a) −−→OA − −→AB
(b) −→OC − −→BC
(c) 3−→BA − 4−→CB
a
b
c
−−→OA − −→AB
−→OC − −→BC
3−→BA − 4−→CB

Dados os vetores ~u = (2,−4), ~v = (−5, 1) e ~w = (−12, 6), determinar a1 e a2 tais que
~w = a1 · ~u + a2 · ~v

~w = a1 · ~u + a2 · ~v

Dados os pontos A(3,−4) e B(−1, 1) e o vetor ~v = (−2, 3), calcular:
(a) (B − A) + 2~v
(b) (A − B) − ~v
(c) B + 2(B − A)
(d) 3~v + 2(A − B)
a
b
c
d
(B − A) + 2~v
(A − B) − ~v
B + 2(B − A)
3~v + 2(A − B)

Encontrar o vértice oposto a B, no paralelogramo ABCD, para:
(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)
a
b
Encontrar o vértice oposto a B, no paralelogramo ABCD, para:
(a) A(−3,−1), B(4, 2) e C(5, 5) (b) A(5, 1), B(7, 3) e C(3, 4)