Question

determine a equação geral do plano que passa pelos pontos A(0,0,0); B(0,3,0);C(0,2,5).​

Answer 1

✅ Após resolver os cálculos, concluímos que a equação geral do referido plano é:

         $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \pi: 15x = 0\:\:\:}}\end{gathered} }$

Sejam os pontos do espaço tridimensional:

                $\Large\begin{cases} A(0, 0, 0)\\B(0, 3, 0)\\C(0, 2, 5)\end{cases}$

Para resolver esta questão devemos:

• Calcular o vetor  diretor "u".

           $\Large\displaystyle\begin{gathered} \vec{u} = \overrightarrow{AB} = B - A = (0, 3, 0) - (0, 0, 0)\end{gathered}$

                $\Large\displaystyle\begin{gathered} = (0 - 0, 3 - 0, 0 - 0) = (0, 3, 0)\end{gathered}$

          $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{u} = (0, 3, 0)\end{gathered}$

• Calcular o vetor diretor "v".

           $\Large\displaystyle\begin{gathered} \vec{v} = \overrightarrow{AC} = C - A = (0, 2, 5) - (0, 0, 0)\end{gathered}$

                $\Large\displaystyle\begin{gathered} = (0 - 0, 2 - 0, 5 - 0) = (0, 2, 5)\end{gathered}$

          $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{v} = (0, 2, 5)\end{gathered}$

• Determinar o vetor normal "n" ao plano. Para isso, devemos calcular o produto vetorial dos vetores diretores.

           $\Large\displaystyle\begin{gathered} \vec{n} = \vec{u}\wedge\vec{v}\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\0 & 3 & 0\\0 & 2 & 5\end{vmatrix}\end{gathered}$  

                 $\Large\displaystyle\begin{gathered} = \begin{vmatrix} 3 & 0\\2 & 5\end{vmatrix}\vec{i} - \begin{vmatrix} 0 & 0\\0 & 5\end{vmatrix}\vec{j} + \begin{vmatrix} 0 & 3\\0 & 2\end{vmatrix}\vec{k}\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = (15 - 0)\vec{i} - (0 - 0)\vec{j} + (0 - 0)\vec{k}\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = 15\vec{i} - 0\vec{j} + 0\vec{k}\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = (15, 0, 0)\end{gathered}$

            $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{n} = (15, 0, 0)\end{gathered}$

• Montar a equação geral do plano. Para isso, devemos utilizar a seguinte fórmula:

          $\large\displaystyle\begin{gathered} X_{n}\cdot X + Y_{n}\cdot Y + Z_{n}\cdot Z = X_{n}\cdot X_{A} + Y_{n}\cdot Y_{A} + Z_{n}\cdot Z_{A}\end{gathered}$

             $\Large\displaystyle\begin{gathered} 15\cdot x + 0\cdot y + 0 \cdot z = 15\cdot0 + 0\cdot 0 + 0\cdot0\end{gathered}$

                                                $\Large\displaystyle\begin{gathered} 15x = 0\end{gathered}$  

✅ Portanto, a equação do plano é:

                                            $\Large\displaystyle\begin{gathered} \pi: 15x = 0\end{gathered}$

         

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$