Question

Encontre os escalares a,b e c que satisfaçam a expressão au+bv+cw=(0,1,4). sabendo que u=(1,4,0), v=(0,-1,2) e w=(3,1,4).

Answer 1

$au+bv+cw =(0,1,4)\\\\a(1,4,0)+b(0,-1,2)+c(3,1,4)=(0,1,4)\\\\(a,4a,0)+(0,-b,2b)+(3c+c+c4)=(0,1,4)\\\\(a+0+3c , 4a-b+c, 0+2b+4c)=(0,1,4)\\\\(a+3c,4a-b+c,2b+4c)=(0,1,4)\\\\\\ \Bmatrix a+3c=0\\4a-b+c=1\\2b+4c=4 \end$

resolvendo o sistema:
vou utilizar a regra de cramer 

$\Bmatrix a+0b+3c=0\\4a-b+c=1\\0a+2b+4c=4 \end$


$D=\left[\begin{array}{ccc}1&0&3\\4&-1&1\\0&2&4\end{array}\right] = 18\\\\\\ D_a=\left[\begin{array}{ccc}0&0&3\\1&-1&1\\4&2&4\end{array}\right] = 18\\\\\\ D_b=\left[\begin{array}{ccc}1&0&3\\4&1&1\\0&4&4\end{array}\right] = 48\\\\\\ D_c=\left[\begin{array}{ccc}1&0&0\\4&-1&1\\0&2&4\end{array}\right] = -6$



$a= \frac{D_a}{D}= \frac{18}{18}=1\\\\ b= \frac{D_b}{D}= \frac{48}{18} = \frac{8}{3}\\\\c= \frac{D_c}{D}= \frac{-6}{18} = \frac{-1}{3}$