Question

dados os vetores u=2i-j e w= -3i, determine t de modo que 3t-(4u-2w)=5.(-t+1/2u-3/4w)

Answer 1

Sendo:

$\vec{u}=2\hat{i}-\hat{j}\\ \vec{w}=-3\hat{i}\\ \vec{t}=a\hat{i}+b\hat{j}$

Devemos encontrar $\vec{t}$ tal que:

$3\vec{t}-(4\vec{u}-2\vec{w})=5\bigg(-\vec{t}+\dfrac{1}{2}\vec{u}-\dfrac{3}{4}\vec{w}\bigg)$

Podemos isolar $\vec{t}$:

$3\vec{t}-4\vec{u}+2\vec{w}=-5\vec{t}+\dfrac{5}{2}\vec{u}-\dfrac{15}{4}\vec{w}\ \to$

$8\vec{t}=\bigg(\dfrac{5}{2}+4\bigg)\vec{u}-\bigg(\dfrac{15}{4}+2\bigg)\vec{w}\ \to\ 8\vec{t}=\dfrac{13}{2}\vec{u}-\dfrac{23}{4}\vec{w}\ \to$

$\boxed{\vec{t}=\dfrac{13}{16}\vec{u}-\dfrac{23}{32}\vec{w}}$

Dessa forma:

$a\hat{i}+b\hat{j}=\dfrac{13}{16}(2\hat{i}-\hat{j})-\dfrac{23}{32}(-3\hat{i})\ \to$

$a\hat{i}+b\hat{j}=\bigg(\dfrac{26}{16}+\dfrac{69}{32}\bigg)\hat{i}-\dfrac{13}{16}\hat{j}\ \therefore\ \boxed{\vec{t}=\dfrac{121}{32}\hat{i}-\dfrac{13}{16}\hat{j}}$