Considerando os estudos sobre produtos entre vetores e dados os vetores u=( -1,3,2), v=(1,5,-2) e w=(-7,3,1) analise as afirmativas, a seguir:
I. u x v = (-16, 0, -8)
II. v x w = (11, 13, 38)
III. u . v = 20
IV. u.(v x w) = 104
V. (v x u) x w = (24, 0, 64)
Quais estão corretas?
a) III, IV e V
b) I, II e III
c) I, IV e V
d) I, II e IV
e) II, III e V
ResolvaMatematica:
Alguém pode me ajudar?
Soluções para a tarefa
Respondido por
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Olá
Alternativa correta, letra D) I, II e IV
Vamos fazer um item de cada vez
I - u x v = (-16, 0, -8 )
Vamos então calcular o produto vetorial entre u e v, para ver se irá resultar no mesmo do enunciado.
![\vec{u}=(-1,3,2)\\\vec{v}=(1,5,-2)\\\\\\ \left[\begin{array}{ccc}i&j&k\\-1&3&2\\1&5&-2\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{-6i+2j-5k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{2j+10i+3k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{-6i+2j-5k-2j-10i-3k}\\\\=\mathsf{-16i+0j-8k}\\\\\\=\boxed{(-16,0,-8)}~~\checkmark~~~~ ~\text{Verdadeira}](https://tex.z-dn.net/?f=%5Cvec%7Bu%7D%3D%28-1%2C3%2C2%29%5C%5C%5Cvec%7Bv%7D%3D%281%2C5%2C-2%29%5C%5C%5C%5C%5C%5C++++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26amp%3Bj%26amp%3Bk%5C%5C-1%26amp%3B3%26amp%3B2%5C%5C1%26amp%3B5%26amp%3B-2%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cunderbrace%7B%28%5Cmathsf%7B-6i%2B2j-5k%7D%29%7D_%7Bdiag.%7Eprincipal%7D%7E%7E-%7E%7E%5Cunderbrace%7B%28%5Cmathsf%7B2j%2B10i%2B3k%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%3D%5Cmathsf%7B-6i%2B2j-5k-2j-10i-3k%7D%5C%5C%5C%5C%3D%5Cmathsf%7B-16i%2B0j-8k%7D%5C%5C%5C%5C%5C%5C%3D%5Cboxed%7B%28-16%2C0%2C-8%29%7D%7E%7E%5Ccheckmark%7E%7E%7E%7E+%7E%5Ctext%7BVerdadeira%7D "\vec{u}=(-1,3,2)\\\vec{v}=(1,5,-2)\\\\\\ \left[\begin{array}{ccc}i&j&k\\-1&3&2\\1&5&-2\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{-6i+2j-5k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{2j+10i+3k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{-6i+2j-5k-2j-10i-3k}\\\\=\mathsf{-16i+0j-8k}\\\\\\=\boxed{(-16,0,-8)}~~\checkmark~~~~ ~\text{Verdadeira}")
Vamos agora ao item II
v x w = (11,13,38)
Calculando o produto vetorial entre v e w
![\vec{v}=(1,5,-2)\\\vec{w}=(-7,3,1)\\\\\\ \left[\begin{array}{ccc}i&j&k\\1&5&-2\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{5i+14j+3k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{j-6i-35k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{5i+14j+3k-j+6i+35k}\\\\=\mathsf{11i+13j+38k}\\\\\\=\boxed{(11,13,38)}~~\checkmark~~~~ ~\text{Verdadeira}](https://tex.z-dn.net/?f=%5Cvec%7Bv%7D%3D%281%2C5%2C-2%29%5C%5C%5Cvec%7Bw%7D%3D%28-7%2C3%2C1%29%5C%5C%5C%5C%5C%5C+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26amp%3Bj%26amp%3Bk%5C%5C1%26amp%3B5%26amp%3B-2%5C%5C-7%26amp%3B3%26amp%3B1%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cunderbrace%7B%28%5Cmathsf%7B5i%2B14j%2B3k%7D%29%7D_%7Bdiag.%7Eprincipal%7D%7E%7E-%7E%7E%5Cunderbrace%7B%28%5Cmathsf%7Bj-6i-35k%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%3D%5Cmathsf%7B5i%2B14j%2B3k-j%2B6i%2B35k%7D%5C%5C%5C%5C%3D%5Cmathsf%7B11i%2B13j%2B38k%7D%5C%5C%5C%5C%5C%5C%3D%5Cboxed%7B%2811%2C13%2C38%29%7D%7E%7E%5Ccheckmark%7E%7E%7E%7E+%7E%5Ctext%7BVerdadeira%7D "\vec{v}=(1,5,-2)\\\vec{w}=(-7,3,1)\\\\\\ \left[\begin{array}{ccc}i&j&k\\1&5&-2\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{5i+14j+3k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{j-6i-35k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{5i+14j+3k-j+6i+35k}\\\\=\mathsf{11i+13j+38k}\\\\\\=\boxed{(11,13,38)}~~\checkmark~~~~ ~\text{Verdadeira}")
Vamos agora ao item III
u.v=20
Calculando o produto escalar entre u e v
Item IV
O produto misto entre u,v, w é 104.
Para calcular esse item, basta montar um matriz 3x3 com os vetores, e em seguida calcular o determinante.
![\vec{u}=( -1,3,2)\\ \vec{v}=(1,5,-2) \\ \vec{w}=(-7,3,1)\\\\\ [\vec{u},\vec{v},\vec{w}] =\left[\begin{array}{ccc}-1&3&2\\1&5&-2\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{-5+42+6})}_{diag.~principal}~~-~~\underbrace{(\mathsf{3+6-70})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{(43)-(-61)}\\\\=\mathsf{43+61}\\\\\\=\boxed{104}~~\checkmark~~~~ ~\text{Verdadeira}](https://tex.z-dn.net/?f=%5Cvec%7Bu%7D%3D%28+-1%2C3%2C2%29%5C%5C+%5Cvec%7Bv%7D%3D%281%2C5%2C-2%29+%5C%5C+%5Cvec%7Bw%7D%3D%28-7%2C3%2C1%29%5C%5C%5C%5C%5C+%5B%5Cvec%7Bu%7D%2C%5Cvec%7Bv%7D%2C%5Cvec%7Bw%7D%5D+%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%26amp%3B3%26amp%3B2%5C%5C1%26amp%3B5%26amp%3B-2%5C%5C-7%26amp%3B3%26amp%3B1%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cunderbrace%7B%28%5Cmathsf%7B-5%2B42%2B6%7D%29%7D_%7Bdiag.%7Eprincipal%7D%7E%7E-%7E%7E%5Cunderbrace%7B%28%5Cmathsf%7B3%2B6-70%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%3D%5Cmathsf%7B%2843%29-%28-61%29%7D%5C%5C%5C%5C%3D%5Cmathsf%7B43%2B61%7D%5C%5C%5C%5C%5C%5C%3D%5Cboxed%7B104%7D%7E%7E%5Ccheckmark%7E%7E%7E%7E+%7E%5Ctext%7BVerdadeira%7D "\vec{u}=( -1,3,2)\\ \vec{v}=(1,5,-2) \\ \vec{w}=(-7,3,1)\\\\\ [\vec{u},\vec{v},\vec{w}] =\left[\begin{array}{ccc}-1&3&2\\1&5&-2\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{-5+42+6})}_{diag.~principal}~~-~~\underbrace{(\mathsf{3+6-70})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{(43)-(-61)}\\\\=\mathsf{43+61}\\\\\\=\boxed{104}~~\checkmark~~~~ ~\text{Verdadeira}")
Agora o item V
Temos que calcular o produto vetorial entre v e u, e em seguida, pegar o vetor resultante e calcular o produto vetorial entre ele e o vetor w
Primeiro então, vamos calcular o produto entre v e u.
![\vec{v}=(1,5,-2)\\\vec{u}=(-1,3,2)\\\\\\ \left[\begin{array}{ccc}i&j&k\\1&5&-2\\-1&3&2\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{10i+2j+3k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{2j-6i-5k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{10i+2j+3k-2j+6i+5k}\\\\=\mathsf{16i+0j+8k}\\\\\\\boxed{v\times u=(16,0,8)}](https://tex.z-dn.net/?f=%5Cvec%7Bv%7D%3D%281%2C5%2C-2%29%5C%5C%5Cvec%7Bu%7D%3D%28-1%2C3%2C2%29%5C%5C%5C%5C%5C%5C+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26amp%3Bj%26amp%3Bk%5C%5C1%26amp%3B5%26amp%3B-2%5C%5C-1%26amp%3B3%26amp%3B2%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cunderbrace%7B%28%5Cmathsf%7B10i%2B2j%2B3k%7D%29%7D_%7Bdiag.%7Eprincipal%7D%7E%7E-%7E%7E%5Cunderbrace%7B%28%5Cmathsf%7B2j-6i-5k%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%3D%5Cmathsf%7B10i%2B2j%2B3k-2j%2B6i%2B5k%7D%5C%5C%5C%5C%3D%5Cmathsf%7B16i%2B0j%2B8k%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7Bv%5Ctimes+u%3D%2816%2C0%2C8%29%7D "\vec{v}=(1,5,-2)\\\vec{u}=(-1,3,2)\\\\\\ \left[\begin{array}{ccc}i&j&k\\1&5&-2\\-1&3&2\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{10i+2j+3k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{2j-6i-5k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{10i+2j+3k-2j+6i+5k}\\\\=\mathsf{16i+0j+8k}\\\\\\\boxed{v\times u=(16,0,8)}")
Agora, pegamos esse vetor que acabamos de encontrar e fazemos o produto vetorial com w
![\vec{v}\times\vec{u}=(16,0,8)\\\vec{w}=(-7,3,1)\\\\\\ \left[\begin{array}{ccc}i&j&k\\16&0&8\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{0i-56j+48k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{16j+24i+0k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{-56j+48k-16j-24i}\\\\=\mathsf{-24i-72j+48k}\\\\\\\boxed{=(-24,-72,48)}~\times ~~~~ ~\text{Falso}](https://tex.z-dn.net/?f=%5Cvec%7Bv%7D%5Ctimes%5Cvec%7Bu%7D%3D%2816%2C0%2C8%29%5C%5C%5Cvec%7Bw%7D%3D%28-7%2C3%2C1%29%5C%5C%5C%5C%5C%5C+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Di%26amp%3Bj%26amp%3Bk%5C%5C16%26amp%3B0%26amp%3B8%5C%5C-7%26amp%3B3%26amp%3B1%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cunderbrace%7B%28%5Cmathsf%7B0i-56j%2B48k%7D%29%7D_%7Bdiag.%7Eprincipal%7D%7E%7E-%7E%7E%5Cunderbrace%7B%28%5Cmathsf%7B16j%2B24i%2B0k%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%3D%5Cmathsf%7B-56j%2B48k-16j-24i%7D%5C%5C%5C%5C%3D%5Cmathsf%7B-24i-72j%2B48k%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%3D%28-24%2C-72%2C48%29%7D%7E%5Ctimes+%7E%7E%7E%7E+%7E%5Ctext%7BFalso%7D "\vec{v}\times\vec{u}=(16,0,8)\\\vec{w}=(-7,3,1)\\\\\\ \left[\begin{array}{ccc}i&j&k\\16&0&8\\-7&3&1\end{array}\right] \\\\\\\mathsf{=\underbrace{(\mathsf{0i-56j+48k})}_{diag.~principal}~~-~~\underbrace{(\mathsf{16j+24i+0k})}_{diag.~secund\'aria}}\\\\\\\\=\mathsf{-56j+48k-16j-24i}\\\\=\mathsf{-24i-72j+48k}\\\\\\\boxed{=(-24,-72,48)}~\times ~~~~ ~\text{Falso}")
Alternativa correta, letra D) I, II e IV
Vamos fazer um item de cada vez
I - u x v = (-16, 0, -8 )
Vamos então calcular o produto vetorial entre u e v, para ver se irá resultar no mesmo do enunciado.
Vamos agora ao item II
v x w = (11,13,38)
Calculando o produto vetorial entre v e w
Vamos agora ao item III
u.v=20
Calculando o produto escalar entre u e v
Item IV
O produto misto entre u,v, w é 104.
Para calcular esse item, basta montar um matriz 3x3 com os vetores, e em seguida calcular o determinante.
Agora o item V
Temos que calcular o produto vetorial entre v e u, e em seguida, pegar o vetor resultante e calcular o produto vetorial entre ele e o vetor w
Primeiro então, vamos calcular o produto entre v e u.
Agora, pegamos esse vetor que acabamos de encontrar e fazemos o produto vetorial com w
cê é fera!
Opa, obrigado! qualquer dúvida, é só perguntar :-)
Só precisava de uma resposta concreta! Faz muito tempo que não mexo com vetores. kkkkkkkk desde o terceiro período que não vejo mais isso, perde a prática.
é verdade, só lembro por que pratico bastante aqui no site. rsrs.
vê isso no começo da faculdade, depois nunca mais.
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Boa tarde
u = (-1,3,2), v = (1,5,-2) , w = (-7,3,1)
I) produto vetorial uxv
i j k i j
-1 3 2 -1 2
1 5 -2 1 5
|uxv| = -6i + 2j - 5k - 3k - 10i - 2j
|uxv| = -16i + 0j - 8k = (-16, 0, - 8) (V)
II) produto vetorial vxw
i j k i j
1 5 -2 1 5
-7 3 1 -7 3
|vxw| = 5i + 14j + 3k + 35k + 6i - j
|vxw| = 11i + 13j + 38k = (11,13,38) (V)
III) produto escalar u.v
u.v = (-1,3,2).(1,5,-2) = -1 + 15 - 4 = 10 (F)
IV) u.(vxw)
(-1,3,2).(11,13,38) = -11 + 39 + 76 = 104 (V)
V) (v x u) x w
(v x u) = -(u x v) = (16, 0, 8)
(16, 0, 8)x(-7 3 1)
i j k i j
16 0 8 16 0
-7 3 1 -7 3
(v x u) x w = 0i - 56j + 48k - 0k - 24i - 16j
(v x u) x w = -24i - 72j + 48k = (-24, -72, 48) (F)
resumindo
I (V)
II) (V)
III) (F)
IV (V)
V (F)
alternativa d) I, II e IV
u = (-1,3,2), v = (1,5,-2) , w = (-7,3,1)
I) produto vetorial uxv
i j k i j
-1 3 2 -1 2
1 5 -2 1 5
|uxv| = -6i + 2j - 5k - 3k - 10i - 2j
|uxv| = -16i + 0j - 8k = (-16, 0, - 8) (V)
II) produto vetorial vxw
i j k i j
1 5 -2 1 5
-7 3 1 -7 3
|vxw| = 5i + 14j + 3k + 35k + 6i - j
|vxw| = 11i + 13j + 38k = (11,13,38) (V)
III) produto escalar u.v
u.v = (-1,3,2).(1,5,-2) = -1 + 15 - 4 = 10 (F)
IV) u.(vxw)
(-1,3,2).(11,13,38) = -11 + 39 + 76 = 104 (V)
V) (v x u) x w
(v x u) = -(u x v) = (16, 0, 8)
(16, 0, 8)x(-7 3 1)
i j k i j
16 0 8 16 0
-7 3 1 -7 3
(v x u) x w = 0i - 56j + 48k - 0k - 24i - 16j
(v x u) x w = -24i - 72j + 48k = (-24, -72, 48) (F)
resumindo
I (V)
II) (V)
III) (F)
IV (V)
V (F)
alternativa d) I, II e IV
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