✅ Após resolver os cálculos, concluímos que a distância entre os referidos pontos é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf D(P_{1}P_{2}) = \frac{2\sqrt{2}}{9}\:u\cdot c\:\:\:}}\end{gathered}$}[/tex]
Sejam os pontos:
[tex]\Large\displaystyle\text{$\begin{gathered} P_{1} = \bigg(-\frac{1}{9},\,0,\,\frac{10}{9}\bigg)\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} P_{2} = \bigg(\frac{1}{9},\,0,\,\frac{8}{9}\bigg)\end{gathered}$}[/tex]
Para calcular a distância entre estes pontos fazemos:
[tex]\large\displaystyle\text{$\begin{gathered} D(P_{1}P_{2}) = \sqrt{(X_{P_{2}} - X_{P_{1}})^{2} + (Y_{P_{2}} - Y_{P_{1}})^{2} + (Z_{P_{2}} - Z_{P_{1}})^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{1}{9} -\bigg(-\frac{1}{9}\bigg)\bigg)^{2} + (0 - 0)^{2} + \bigg(\frac{8}{9} - \frac{10}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{1}{9} + \frac{1}{9}\bigg)^{2} + 0^{2} + \bigg(\frac{8}{9} - \frac{10}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{2}{9}\bigg)^{2} + \bigg(-\frac{2}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{2^{2}}{9^{2}} + \frac{(-2)^{2}}{9^{2}}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{4}{81} + \frac{4}{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{8}{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \frac{\sqrt{8}}{\sqrt{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \frac{2\sqrt{2}}{9}\end{gathered}$}[/tex]
✅ Portanto, a distância procurada é:
[tex]\Large\displaystyle\text{$\begin{gathered} D(P_{1}P_{2}) = \frac{2\sqrt{2}}{9}\:u\cdot c\end{gathered}$}[/tex]
Saiba mais:
Solução gráfica (figura):
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✅ Após resolver os cálculos, concluímos que a distância entre os referidos pontos é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf D(P_{1}P_{2}) = \frac{2\sqrt{2}}{9}\:u\cdot c\:\:\:}}\end{gathered}$}[/tex]
Sejam os pontos:
[tex]\Large\displaystyle\text{$\begin{gathered} P_{1} = \bigg(-\frac{1}{9},\,0,\,\frac{10}{9}\bigg)\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} P_{2} = \bigg(\frac{1}{9},\,0,\,\frac{8}{9}\bigg)\end{gathered}$}[/tex]
Para calcular a distância entre estes pontos fazemos:
[tex]\large\displaystyle\text{$\begin{gathered} D(P_{1}P_{2}) = \sqrt{(X_{P_{2}} - X_{P_{1}})^{2} + (Y_{P_{2}} - Y_{P_{1}})^{2} + (Z_{P_{2}} - Z_{P_{1}})^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{1}{9} -\bigg(-\frac{1}{9}\bigg)\bigg)^{2} + (0 - 0)^{2} + \bigg(\frac{8}{9} - \frac{10}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{1}{9} + \frac{1}{9}\bigg)^{2} + 0^{2} + \bigg(\frac{8}{9} - \frac{10}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\bigg(\frac{2}{9}\bigg)^{2} + \bigg(-\frac{2}{9}\bigg)^{2}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{2^{2}}{9^{2}} + \frac{(-2)^{2}}{9^{2}}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{4}{81} + \frac{4}{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \sqrt{\frac{8}{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \frac{\sqrt{8}}{\sqrt{81}}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} = \frac{2\sqrt{2}}{9}\end{gathered}$}[/tex]
✅ Portanto, a distância procurada é:
[tex]\Large\displaystyle\text{$\begin{gathered} D(P_{1}P_{2}) = \frac{2\sqrt{2}}{9}\:u\cdot c\end{gathered}$}[/tex]
Saiba mais:
Solução gráfica (figura):