Question

Find the vector equation of the plane passing through the points (1, 1, 1), (1, −1, 1) and (−7, −3, −5).

Answer 1

\[ \text{ Let } A(1, 1, 1),B(1, -1, 1) \text{ and } C(-7, -3, -5) \text{ be the coordinates } .\]
\[\text{ The required plane passes through the point} A(1, 1, 1) \text{ whose position vector is }  \vec{a} = \hat{i} + \hat{j} + \hat{k} \text{ and is normal to the vector }  \vec{n} \text{ given by } \]
\[ \vec{n} = \vec{AB} \times \vec{AC .} \]
\[\text{ Clearly } , \vec{AB} = \vec{OB} - \vec{OA} = \left( \hat{i}  - \hat{j}  + \hat{k}  \right) - \left( \hat{i}  + \hat{j}  + \hat{k} \right) = 0 \hat{i}  - 2 \hat{j} + 0 \hat{k}  \]
\[ \vec{AC} = \vec{OC} - \vec{OA} = \left( - 7 \hat{i}  - 3 \hat{j} - 5 \hat{k}  \right) - \left( \hat{i}  + \hat{j}  + \hat{k} \right) = - 8 \hat{i}  - 4 \hat{j}  - 6 \hat{k} \]
\[ \vec{n} = \vec{AB} ×  \vec{AC} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\0 & - 2 & 0 \\ - 8 & - 4 & - 6\end{vmatrix} = 12 \hat{i} + 0 \hat{j} - 16 \hat{k}  \]
\[\text{ The vector equation of the required plane is }\]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( 12 \hat{i} + 0 \hat{j} - 16 \hat{k} \right) = \left( \hat{i} + \hat{j} + \hat{k} \right) . \left( 12 \hat{i} + 0 \hat{j} - 16 \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left[ 4 \left( 3 \hat{i} - 4 \hat{k} \right) \right] = 12 + 0 - 16\]
\[ \Rightarrow \vec{r} . \left[ 4 \left( 3 \hat{i} - 4 \hat{k}  \right) \right] = - 4\]
\[ \Rightarrow \vec{r} . \left( 3 \hat{i}  - 4 \hat{k}  \right) = - 1\]
\[ \Rightarrow \vec{r} . \left( 3 \hat{i}  - 4 \hat{k} \right) + 1 = 0\]