Question

Determinar o vetor v,paralelo ao vetor u=(2,-1,3),tal que u.v=-18

Answer 1

Seja "x" um escalar, se o vetor V é paralelo a U, então:

$\vec{V}~=~\Vec{U}.\,x\\\\\\\vec{V}~=~(2~,\,-1~,~3).\,x\\\\\\\boxed{\vec{V}~=~(2x~,\,-x~,~3x)}$

Sendo assim, substituindo os vetores na equação dada:

$\vec{U}.\,\vec{V}~=~-18\\\\\\(2~,\,-1~,~3).(2x~,\,-x~,~3x)~=~-18\\\\\\2~.~2x~+~(-1)~.~(-x)~+~3~.~3x~=~-18\\\\\\4x~+~x~+~9x~=~-18\\\\\\14x~=~-18\\\\\\x~=~\frac{-18}{14}\\\\\\\boxed{x~=~-\frac{9}{7}}$

Dessa forma, o vetor V fica:

$\vec{V}~=~(2x~,\,-x~,~3x)\\\\\\\vec{V}~=~\left(2\,.\,\right(-\frac{9}{7}\left)~,\,-\right(-\frac{9}{7}\left)~,~3\,.\,\right(-\frac{9}{7}\left)\right)\\\\\\\boxed{\vec{V}~=~\left(-\frac{18}{7}~,~\frac{9}{7}~,\,-\frac{27}{7}\right)}$