Question

Show that the following planes are at right angles.

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

 

Answer 1

` \text{ We know that the planes }  \vec{r} . \vec{n_1} = d_1 , \vec{r} . \vec{n_2} = d_2 \text{ are perpendicular to each other only if } \vec{n_1} . \vec{n_2} =0.`
\[\text{ Here } , \vec{n_1} = 2 \hat{i} - \hat{j} + \hat{k} ; \vec{n_2} = - \hat{i} - \hat{j} + \hat{k}  \]
\[\text{ Now } ,\]
\[ \vec{n_1} . \vec{n_2} = \left( 2 \hat{i}  - \hat{j}  + \hat{k}  \right) . \left( - \hat{i}  - \hat{j}  + \hat{k}  \right) = - 2 + 1 + 1 = 0\]
\[\text{ So, the given planes are perpendicular } .\]