Question

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

Answer 1

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.

$\vec{u}=(4,-1,-2)\\\vec{v}=(-1,-3,0)\\\vec{w}=(2,7,k)\\\\\\ \left[\begin{array}{ccc}4&-1&-2\\-1&-3&0\\2&7&k\end{array}\right] =0\\\\\\\mathsf{\underbrace{(\mathsf{-12k+0+14})}_{diag.~principal}-\underbrace{(\mathsf{k+0+12})}_{diag.~secund\'aria}}=0\\\\\\\mathsf{-12k-k+14-12=0}\\\\\mathsf{-13k+2=0}\\\\\mathsf{-13k=-2}\\\\\mathsf{k= \frac{-2}{-13}}\\\\\boxed{\mathsf{k= \frac{2}{13} }}~~~\Longrightarrow~~~\text{Letra B}$