Question

Me ajudem com os vetores ?

Dados os pontos A(-1,0,2),B(-4,1,1),C(0,1,3) determinar o vetor x tal que 2x - AB = x + (BC.AB)AC

Answer 1

$A(-1;0;2)\\\\B(-4;1;1)\\\\C(0;1;3)$

AB = B-A 
AC= C-A
BC = C-B
$AB=(-3;1;-1)\\\\AC=(1;1;1)\\\\BC=(4;0;2)$


::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
$2X-AB=X+(BC*AB)*AC\\\\2X=AB+X+(BC*AB)*AC\\\\2X-X=AB+(BC*AB)*AC\\\\\ \boxed{X=AB+(BC*AB)*AC}$

primeiro resolvendo dentro do parentes (BC*AC) utilizando o produto escalar
$(BC*AC)=(4;0;2)*(-3;1;-1)\\\\(BC*AC)=(4*-3)+(0*1)+(2*-1)\\\\(BC*AC)=-12+0-2\\\\\boxed{BC*AC=-14}$

agora temos 
$X=AB+(BC*AB)*AC}\\\\X=AB+(-14)*AC}$

resolvendo a multiplicação...multiplicando um numero por um vetor ..
$-14*AC\\\\-14*(1;1;1)\\\\(-14;-14;-14)$

temos
$X=AB+(-14)*AC\\\\X=AB+(-14;-14;-14)$

somando com AB
$AB+(-14-14;-14)\\\\(-3;1;-1)+(-14;-14;-14)\\\\(-17;-13;-15)$

$X=AB+(-14;-14;-14)\\\\\\\boxed{X=(-17;-13;-15)}$