Question

sabendo que o plano pi terminado pelo ponto a(2,-1,3) b(3,2,-4) e c (2,2,5) assinale a alternativa que representa à equação geral de pi

Answer 1

✅ Após resolver os cálculos, concluímos que a equação geral do plano que contém os três referidos pontos é:

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \pi: 27x - 2y + 3z - 65 = 0 \:\:\:}}\end{gathered} }$

Sejam os pontos:

                  $\Large\begin{cases} A(2, -1, 3)\\B(3, 2, -4)\\C(2, 2, 5)\end{cases}$

Para resolver esta questão devemos:

• Calcular o vetor diretor "u":

           $\Large\displaystyle\begin{gathered} \vec{u} = \overrightarrow{AB}\end{gathered}$

               $\Large\displaystyle\begin{gathered} = B - A\end{gathered}$

               $\Large\displaystyle\begin{gathered} = (3, 2, -4) - (2, -1, 3)\end{gathered}$

                $\Large\displaystyle\begin{gathered} = (3 - 2, 2 - (-1), -4 - 3)\end{gathered}$

                $\Large\displaystyle\begin{gathered} = (1, 3, -7)\end{gathered}$

                $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{u} = (1, 3, -7)\end{gathered}$

• Calcular o vetor diretor "v":

            $\Large\displaystyle\begin{gathered} \vec{v} = \overrightarrow{AC}\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = C - A\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = (2, 2, 5) - (2, -1, 3)\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = (2-2,2 - (-1), 5 - 3)\end{gathered}$

                 $\Large\displaystyle\begin{gathered} = (0, 3, 2)\end{gathered}$

                 $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{v} = (0, 3, 2)\end{gathered}$

• Calcular o vetor "w" que será o produto vetorial entre "u" e "v":

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

                  $\Large\displaystyle\begin{gathered} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\1 & 3 & -7\\0 & 3 & 2\end{vmatrix}\end{gathered}$

                  $\Large\displaystyle\begin{gathered} = \begin{vmatrix} 3 & -7\\3 & 2\end{vmatrix}\cdot\vec{i} - \begin{vmatrix}1 & -7\\0 & 2 \end{vmatrix}\cdot\vec{j} + \begin{vmatrix}1 & 3\\0 & 3 \end{vmatrix}\cdot\vec{k}\end{gathered}$

                  $\Large\displaystyle\begin{gathered} = (6 + 21)\vec{i} - (2 + 0)\vec{j} + (3 - 0)\vec{k}\end{gathered}$

                  $\Large\displaystyle\begin{gathered} = 27\vec{i} - 2\vec{j} + 3\vec{k}\end{gathered}$

                   $\Large\displaystyle\begin{gathered} = (27, -2, 3)\end{gathered}$

                   $\Large\displaystyle\begin{gathered} \therefore\:\:\:\vec{w} = (27, -2, 3)\end{gathered}$

• Montar a equação geral da reta: Para isso, devemos utilizar a seguinte fórmula:

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

        Substituindo as coordenadas do ponto "A" bem como as componentes do vetor "w", temos:

        $\Large\displaystyle\begin{gathered} 27\cdot x + (-2)\cdot y + 3\cdot z = 27\cdot2 + (-2)\cdot(-1) + 3\cdot 3\end{gathered}$

                           $\Large\displaystyle\begin{gathered} 27x - 2y + 3z = 54 + 2 + 9\end{gathered}$

                           $\Large\displaystyle\begin{gathered} 27x - 2y + 3z = 65\end{gathered}$

                $\Large\displaystyle\begin{gathered} 27x - 2y + 3z - 65 = 0\end{gathered}$

✅ Portanto, a equação do plano é:

  $\Large\displaystyle\begin{gathered} \pi: 27x - 2y + 3z - 65 = 0\end{gathered}$

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

Saiba mais:

1. https://brainly.com.br/tarefa/16331001
2. https://brainly.com.br/tarefa/29011448
3. https://brainly.com.br/tarefa/12563022
4. https://brainly.com.br/tarefa/12723145
5. https://brainly.com.br/tarefa/27973259
6. https://brainly.com.br/tarefa/16983186
7. https://brainly.com.br/tarefa/6019002
8. https://brainly.com.br/tarefa/8122459
9. https://brainly.com.br/tarefa/17117351
10. https://brainly.com.br/tarefa/52598208

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