Question

escreva o vetor (7,-1), como a soma e dois vetores, um paralelo ao vetor (1,-1) e outro pararelo ao vetor (1,1)

Answer 1

Queremos encontrar dois vetores $\overrightarrow{\mathbf{u}}$ e $\overrightarrow{\mathbf{v}},$ de forma que

$\bullet\;\;(7,\,-1)=\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{v}}\\\\ \bullet\;\;\overrightarrow{\mathbf{u}}=\alpha(1,\,-1)\\\\ \bullet\;\;\overrightarrow{\mathbf{v}}=\beta(1,\,1)$

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Encontrar $\alpha$ e $\beta$ convenientes:

$(7,\,-1)=\alpha(1,\,-1)+\beta(1,\,1)\\\\ (7,\,-1)=(\alpha,\,-\alpha)+(\beta,\,\beta)\\\\ (7,\,-1)=(\alpha+\beta,\,-\alpha+\beta)\\\\\\ \left\{\!\begin{array}{rcrc} \alpha+\beta&\!\!=\!\!&7&~~~~\mathbf{(i)}\\ -\alpha+\beta&\!\!=\!\!&-1&~~~~\mathbf{(ii)} \end{array} \right.$


Somando-se as equações $\mathbf{(i)}$ e $\mathbf{(ii)}$ membro a membro, obtemos

$\alpha+\beta+(-\alpha+\beta)=7-1\\\\ \beta+\beta=6\\\\ 2\beta=6\\\\ \boxed{\begin{array}{c}\beta=3 \end{array}}$


Encontrando $\alpha:$

$\alpha=7-\beta\\\\ \alpha=7-3\\\\ \boxed{\begin{array}{c}\alpha=4 \end{array}}$

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Os vetores $\overrightarrow{\mathbf{u}}$ e $\overrightarrow{\mathbf{v}}$ são

$\overrightarrow{\mathbf{u}}=4(1,\,-1)=(4,\,-4)\\\\ \overrightarrow{\mathbf{v}}=3(1,\,1)=(3,\,3)$


De modo que

$\boxed{\begin{array}{c}(7,\,-1)=(4,\,-4)+(3,\,3) \end{array}}$


Bons estudos! :-)