Question

Show that the normals to the following pairs of planes are perpendicular to each other.

\[\vec{r} \cdot \left( 2 \hat{i}  - \hat{j}  + 3 \hat{k}  \right) = 5 \text{ and }  \vec{r} \cdot \left( 2 \hat{i}  - 2 \hat{j}  - 2 \hat{k}  \right) = 5\]

Answer 1

`  \text{ Let } \vec{n_1} \text{ and }  \vec{n_2} \text{ be the vectors which are normals to the planes } \vec{r} .\left( 2 \hat{i}  - \hat{j}  + 3 \hat{k}  \right)= 5 \text{ and } \vec{r} .\left( 2 \hat{i}  - 2 \hat{j}  - 2 \hat{k}  \right)= 5 \text{ respectively }. `
\[\text{ The given equations of the planes are }\]
\[ \vec{r} .\left( \text{ 2 }\hat{i} - \hat{j} + \text{  3 }\hat{k} \right)= 5 ; \vec{r} .\left( \text{ 2 } \hat{i} - \text{ 2 } \hat{j} - \text{ 2 }\hat{k} \right)= 5\]
\[ \Rightarrow \vec{n_1} = \left( \text{ 2 }\hat{i} - \hat{j} + \text{ 3 }\hat{k} \right); \vec{n_2} = \left( \text{ 2 } \hat{i} - \text{ 2 } \hat{j} - \text{  2 } \hat{k} \right)\]
\[Now, \vec{n_1} . \vec{n_2} = \left( \text{ 2 }\hat{i} - \hat{j} + \text{ 3 }\hat{k} \right) . \left( \text{ 2 }\hat{i} - \text{  2 } \hat{j} - \text{ 2 }\hat{k} \right) = 4 + 2 - 6 = 0\]
\[\text{ So, the normals to the given planes are perpendicular to each other } .\]