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

Respostas

respondido por: solkarped

✅ 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".

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

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

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

Calcular o vetor diretor "v".

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

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

$\Large\displaystyle\text{$\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.

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

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

$\Large\displaystyle\text{$\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}$}$

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

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

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

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

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

$\large\displaystyle\text{$\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}$}$