Question

dados os vetores u=(1,1,-1) e v= (2,-1,1) e w (x,0,x), Tal que u=wxv

Answer 1

Dados três vetores

$\overrightarrow{\mathbf{u}}=(1,\,1,\,-1)\,,~~\overrightarrow{\mathbf{v}}=(2,\,-1,\,1)~\text{ e }~\overrightarrow{\mathbf{w}}=(x,\,0,\,x)$

encontrar $x\in\mathbb{R}$ tal que

$\overrightarrow{\mathbf{u}}=\overrightarrow{\mathbf{w}}\times \overrightarrow{\mathbf{v}}\\\\\\\ (1,\,1,\,-1)=(x,\,0,\,x)\times (2,\,-1,\,1)\\\\\\ (1,\,1,\,-1)=\det\!\left[\begin{array}{ccc} \overrightarrow{\mathbf{i}}&\overrightarrow{\mathbf{j}}&\overrightarrow{\mathbf{k}}\\ x&0&x\\ 2&-1&1 \end{array}\right ]\\\\\\ (1,\,1,\,-1)=(0\cdot 1-(-1)\cdot x)\overrightarrow{\mathbf{i}}-(x\cdot 1-2\cdot x)\overrightarrow{\mathbf{j}}+(x\cdot (-1)-2\cdot 0)\overrightarrow{\mathbf{k}}\\\\ (1,\,1,\,-1)=x\overrightarrow{\mathbf{i}}+x\overrightarrow{\mathbf{j}}-x\overrightarrow{\mathbf{k}}\\\\ (1,\,1,\,-1)=(x,\,x,\,-x)$


Obviamente, pela última igualdade acima, devemos ter

$\boxed{\begin{array}{c}x=1 \end{array}}$