sejam os vetores U=(4,-1,-2), V=(-1.-3,0) e W=(2,7,K) Três vetroes linearmente dependentes, o valor de k é:
a) K=5/13
b) K=2/13
c) K=7/11
d) K=2/11
e) K=13/2
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Olá
Alternativa correta, letra B)
Se 3 vetores são linearmente dependentes, então o determinante entre esses 3 vetores é igual a zero.
Então, vamos criar uma matriz 3x3 com os vetores e calcular o determinante e igualar a zero.
![\vec{u}=(4,-1,-2)\\\vec{v}=(-1,-3,0)\\\vec{w}=(2,7,k)\\\\\\ \left[\begin{array}{ccc}4&-1&-2\\-1&-3&0\\2&7&k\end{array}\right] =0\\\\\\\mathsf{\underbrace{(\mathsf{-12k+0+14})}_{diag.~principal}-\underbrace{(\mathsf{k+0+12})}_{diag.~secund\'aria}}=0\\\\\\\mathsf{-12k-k+14-12=0}\\\\\mathsf{-13k+2=0}\\\\\mathsf{-13k=-2}\\\\\mathsf{k= \frac{-2}{-13}}\\\\\boxed{\mathsf{k= \frac{2}{13} }}~~~\Longrightarrow~~~\text{Letra B}](https://tex.z-dn.net/?f=%5Cvec%7Bu%7D%3D%284%2C-1%2C-2%29%5C%5C%5Cvec%7Bv%7D%3D%28-1%2C-3%2C0%29%5C%5C%5Cvec%7Bw%7D%3D%282%2C7%2Ck%29%5C%5C%5C%5C%5C%5C++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D4%26amp%3B-1%26amp%3B-2%5C%5C-1%26amp%3B-3%26amp%3B0%5C%5C2%26amp%3B7%26amp%3Bk%5Cend%7Barray%7D%5Cright%5D+%3D0%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%5Cunderbrace%7B%28%5Cmathsf%7B-12k%2B0%2B14%7D%29%7D_%7Bdiag.%7Eprincipal%7D-%5Cunderbrace%7B%28%5Cmathsf%7Bk%2B0%2B12%7D%29%7D_%7Bdiag.%7Esecund%5C%27aria%7D%7D%3D0%5C%5C%5C%5C%5C%5C%5Cmathsf%7B-12k-k%2B14-12%3D0%7D%5C%5C%5C%5C%5Cmathsf%7B-13k%2B2%3D0%7D%5C%5C%5C%5C%5Cmathsf%7B-13k%3D-2%7D%5C%5C%5C%5C%5Cmathsf%7Bk%3D+%5Cfrac%7B-2%7D%7B-13%7D%7D%5C%5C%5C%5C%5Cboxed%7B%5Cmathsf%7Bk%3D+%5Cfrac%7B2%7D%7B13%7D++%7D%7D%7E%7E%7E%5CLongrightarrow%7E%7E%7E%5Ctext%7BLetra+B%7D "\vec{u}=(4,-1,-2)\\\vec{v}=(-1,-3,0)\\\vec{w}=(2,7,k)\\\\\\ \left[\begin{array}{ccc}4&-1&-2\\-1&-3&0\\2&7&k\end{array}\right] =0\\\\\\\mathsf{\underbrace{(\mathsf{-12k+0+14})}_{diag.~principal}-\underbrace{(\mathsf{k+0+12})}_{diag.~secund\'aria}}=0\\\\\\\mathsf{-12k-k+14-12=0}\\\\\mathsf{-13k+2=0}\\\\\mathsf{-13k=-2}\\\\\mathsf{k= \frac{-2}{-13}}\\\\\boxed{\mathsf{k= \frac{2}{13} }}~~~\Longrightarrow~~~\text{Letra B}")
Alternativa correta, letra B)
Se 3 vetores são linearmente dependentes, então o determinante entre esses 3 vetores é igual a zero.
Então, vamos criar uma matriz 3x3 com os vetores e calcular o determinante e igualar a zero.
francissvitor:
muito obrigado, certinha.
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