✅ Após resolver os cálculos, concluímos que a área do paralelogramo é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf S = 2\sqrt{62}\:u.\,a.\:\:\:}}\end{gathered}$}[/tex]
Sejam os vetores:
[tex]\Large\begin{cases} \vec{u} = (1, 1, 3)\\\vec{v} = (-4, 2, 2)\end{cases}[/tex]
Para obtermos a área "S" do paralelogramo a partir de seus vetores diretores devemos calcular o módulo do produto vetorial dos vetores diretores, ou seja:
[tex]\Large\displaystyle\text{$\begin{gathered} S = \parallel\vec{u}\wedge\vec{v}\parallel\end{gathered}$}[/tex]
Para resolver esta questão, devemos:
[tex]\Large\displaystyle\text{$\begin{gathered} \vec{u}\wedge\vec{v}= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\1 & 1 & 3\\-4 & 2 & 2\end{vmatrix}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \begin{vmatrix} 1 & 3\\2 & 2\end{vmatrix}\vec{i} - \begin{vmatrix} 1 & 3\\-4 & 2\end{vmatrix}\vec{j} + \begin{vmatrix} 1 & 1\\-4 & 2\end{vmatrix}\vec{k}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = (2 - 6)\vec{i} - (2 + 12)\vec{j} + (2 + 4)\vec{k}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = -4\,\vec{i} - 14\,\vec{j} + 6\,\vec{k}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = (-4,-14,6)\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \therefore\:\:\:\vec{u}\wedge\vec{v} = (-4,-14, 6)\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \parallel \vec{u}\wedge\vec{v}\parallel = \sqrt{(-4)^{2} + (-14)^{2} + 6^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{16 + 196 + 36}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{248}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = 2\sqrt{62}\end{gathered}$}[/tex]
✅ Portanto, a área do paralelogramo é:
[tex]\Large\displaystyle\text{$\begin{gathered} S = 2\sqrt{62}\:u.\,a.\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
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