Question

O vetor v⃗ =(4,3,−6) em R3 pode ser escrito como combinação linear dos vetores: e⃗ 1=(1,−3,2) e e⃗ 2=(2,4,−1) ?

Answer 1

Bom dia Carol!

Solução!

$Sendo~~a,b~~\in\mathbb{R}^{3}\\\\\\\ v_{1}(1,-3,2)\\\\\\\ v_{2}(2,4-1)\\\\\\\ v=(4,3,-6)\\\\\\\\\\ a(1,-3,2)+b(2,4,-1)=(4,3,-6)\\\\\\ \begin{cases} a+2b=4\\ -3a+4b=3\\ 2a-b=-6 \end{cases}\\\\\\ Fazendo!\\\\\\ -b=-6-2a.(-1)\\\\\ \boxed{b=6+2a}\\\\\\ Substituindo!\\\\\ a+2b=4\\\\\ a+2(6+2a)=4\\\\\ a+12+4a=4\\\\\\ 5a=4-12\\\\\ 5a=-8\\\\\ \boxed{a= -\frac{8}{5}}$


$b=6+2a\\\\\ b=6+2(- \frac{8}{5})\\\\\ b=6- \frac{16}{5}\\\\\\ b= \frac{30-16}{5}\\\\\ \boxed{b= \frac{14}{5}}$


$\boxed{a=- \frac{8}{5}~~b= \frac{14}{5}}$


$- \frac{8}{5}(1,-3,2)+ \frac{14}{5}(2,4,-1)=(4,3,-6)\\\\\\ ( -\frac{8}{5}, \frac{24}{5}, -\frac{16}{5})+( \frac{28}{5}, \frac{56}{5}, \frac{14}{5})=(4,3,-6)\\\\\\ ( \frac{8-28}{5}, \frac{24+56}{5}, \frac{-16+14}{5})=(4,3-6)\\\\\\ \boxed{(4,16, -\frac{2}{5}) \neq (4,3,-6)}$


$\boxed{Resposta:~~N\~ao}$


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