Question

10) Calcule a distância distância do ponto A = (2,1, −2) o plano π: 2x − y + z = −10

Answer 1

✅ Após resolver os cálculos, concluímos que a distância entre o ponto e a reta dada é:

         $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf d_{\overline{A\pi}} = \frac{11\sqrt{6}}{6}\,u.\,c.\:\:\:}}\end{gathered} }$

Sejam os dados:

         $\Large\begin{cases} A = (2, 1, -2)\\\pi: 2x - y + z = -10\end{cases}$

Podemos reescrever os dados como:

         $\Large\begin{cases} A = (2, 1, -2)\\\pi: 2x - y + z + 10 = 0\end{cases}$

Para calcular a distância entre o ponto "A" e o plano "π", devemos utilizar a seguinte fórmula:

$\Large\displaystyle\begin{gathered} \bf(I)\end{gathered}$         $\Large\displaystyle\text{ \begin{gathered} D_{\overline{A\pi}} = \frac{|a_{\pi}x_{A} + b_{\pi}y_{A} + c_{\pi}z_{A} + d_{\pi}|}{\sqrt{a_{\pi}^{2} + b_{\pi}^{2} + c_{\pi}^{2}}}\end{gathered} }$

Substituindo os valores na equação "I", temos:

      $\Large\displaystyle\text{ \begin{gathered} d_{\overline{A\pi}} = \frac{|2\cdot2 + (-1)\cdot1 + 1\cdot(-2) + 10|}{\sqrt{2^{2} + (-1)^{2} + 1^{2}}}\end{gathered} }$

               $\Large\displaystyle\text{ \begin{gathered} = \frac{|4 - 1 - 2 + 10|}{\sqrt{4 + 1 + 1}}\end{gathered} }$

               $\Large\displaystyle\text{ \begin{gathered} = \frac{|11|}{\sqrt{6}}\end{gathered} }$

               $\Large\displaystyle\text{ \begin{gathered} = \frac{11}{\sqrt{6}}\end{gathered} }$

               $\Large\displaystyle\text{ \begin{gathered} = \frac{11}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}}\end{gathered} }$

               $\Large\displaystyle\text{ \begin{gathered} = \frac{11\sqrt{6}}{6}\end{gathered} }$

✅ Portanto, a distância entre o ponto e a reta é:

      $\Large\displaystyle\text{ \begin{gathered} d_{\overline{A\pi}} = \frac{11\sqrt{6}}{6}\,u.\,c.\end{gathered} }$

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

Saiba mais:

1. https://brainly.com.br/tarefa/49478382
2. https://brainly.com.br/tarefa/52627212

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