Question

4. Vetor é dito unitário quando seu módulo é igual ao valor de 1. Calcule o vetor unitário do vetor u (4, 2, 6)

Answer 1

➜ O vetor unitário procurado é $\hat{u} =\left(\frac{2}{\sqrt{14}} ,\frac{1}{\sqrt{14}} ,\frac{3}{\sqrt{14}}\right)$

☞ Módulo do vetor $\vec{u}=(a,b,c)$:

$\Large\boxed{||\vec{u} ||=\sqrt{a^{2} +b^{2} +c^{2}}}$

☞ O versor (vetor unitário, i.e, módulo igual a 1) de $\vec{u}=(a,b,c)$ é dado por:

$\Large\boxed{\hat{u} =\frac{\vec{u}}{||\vec{u} ||}}$

Temos $\vec{u}=(4,2,6)$, então o vetor unitário de u é:

$\begin{array}{l}\hat{u} =\frac{( 4,2,6)}{\sqrt{4^{2} +2^{2} +6^{2}}} =\\\\=\frac{1}{\sqrt{56}} \cdotp ( 4,2,6)\\\\=\frac{1}{2\sqrt{14}} \cdotp ( 4,2,6)\\\\=\left(\frac{4}{2\sqrt{14}} ,\frac{2}{2\sqrt{14}} ,\frac{6}{2\sqrt{14}}\right)\\\\=\left(\frac{2}{\sqrt{14}} ,\frac{1}{\sqrt{14}} ,\frac{3}{\sqrt{14}}\right)\end{array}$

∵ O versor de $\vec{u}=(4,2,6)$ é $\hat{u} =\left(\frac{2}{\sqrt{14}} ,\frac{1}{\sqrt{14}} ,\frac{3}{\sqrt{14}}\right)$.

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Answer 2

✅ Após ter resolvido todos os cálculos, concluímos que o vetor unitário - vetor normalizado - é:

    $\Large\displaystyle\text{ \begin{gathered}\hat{u} = \Bigg(\frac{\sqrt{14}}{7}, \frac{\sqrt{14}}{14}, \frac{3\sqrt{14}}{14} \Bigg) \end{gathered} }$

Seja o vetor:

       $\Large\displaystyle\begin{gathered}\vec{u} = (4, 2, 6) \end{gathered}$

Para calcular o vetor unitário de "u", basta normaliza-lo, ou seja:

      $\Large\displaystyle\text{ \begin{gathered}\hat{u} = \frac{\vec{u}}{\|\vec{u}\|} \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \frac{(4, 2, 6)}{\sqrt{4^{2} + 2^{2} + 6^{2}}} \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \frac{(4, 2, 6)}{\sqrt{16 + 4 + 36}} \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \frac{(4, 2, 6)}{\sqrt{56}} \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{4}{\sqrt{56}}, \frac{2}{\sqrt{56}}, \frac{6}{\sqrt{56}} \Bigg) \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{4}{2\sqrt{14}}, \frac{2}{2\sqrt{14}}, \frac{6}{2\sqrt{14}} \Bigg) \end{gathered} }$

           $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{2}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{3}{\sqrt{14}} \Bigg) \end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{\sqrt{14}}{7}, \frac{\sqrt{14}}{14}, \frac{3\sqrt{14}}{14} \Bigg) \end{gathered} }$

✅ Portanto, o vetor unitário é:

             $\Large\displaystyle\text{ \begin{gathered}\hat{u} = \Bigg(\frac{\sqrt{14}}{7}, \frac{\sqrt{14}}{14}, \frac{3\sqrt{14}}{14} \Bigg) \end{gathered} }$

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