Question

Os vetores foram definidos como u=(2a-1,2,3) v=(0,2,a) e w=(3,4,2). Utilizando os seus conhecimentos, determine "a" de modo que u.v=(u-v).w

Answer 1

$\mathtt{\overrightarrow{\mathtt{u}}=(2a-1,\,2,\,3)}\\\\ \mathtt{\overrightarrow{\mathtt{v}}=(0,\,2,\,a)}\\\\ \mathtt{\overrightarrow{\mathtt{w}}=(3,\,4,\,2)}.$

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Queremos que

$\mathtt{\overrightarrow{\mathtt{u}}\cdot \overrightarrow{\mathtt{v}}=(\overrightarrow{\mathtt{u}}-\overrightarrow{\mathtt{v}})\cdot \overrightarrow{\mathtt{w}}}\\\\ \mathtt{(2a-1,\,2,\,3)\cdot (0,\,2,\,a)=\left[(2a-1,\,2,\,3)-(0,\,2,\,a)\right]\cdot (3,\,4,\,2)}\\\\ \mathtt{(2a-1)\cdot 0+2\cdot 2+3\cdot a=(2a-1-0,\,2-2,\,3-a)\cdot (3,\,4,\,2)}\\\\ \mathtt{4+3a=(2a-1,\,0,\,3-a)\cdot (3,\,4,\,2)}\\\\ \mathtt{4+3a=(2a-1)\cdot 3+0\cdot 4+(3-a)\cdot 2}\\\\ \mathtt{4+3a=6a-3+6-2a}\\\\ \mathtt{4+3a=4a+3}$

$\mathtt{4+3a=4a+3}\\\\ \mathtt{3a-4a=3-4}\\\\ \mathtt{-a=-1}\\\\ \boxed{\begin{array}{c}{\mathtt{a=1}} \end{array}}$


Dúvidas? Comente.


Bons estudos! :-)