Question

Determine the value of λ for which the following planes are perpendicular to each other.

\[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j} + 3 \hat{k} \right) = 7 \text{ and }  \vec{r} \cdot \left( \lambda \hat{i} + 2 \hat{j}  - 7 \hat{k}  \right) = 26\]

 

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.`

\[\text{ Here} , \vec{n_1} = \hat{i}| + 2 \hat{j} + 3 \hat{k} ; \vec{n_2} = \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \]

\[\text{ The given planes are perpendicular.} \]

\[ \Rightarrow \vec{n_1} . \vec{n_2} = 0\]

\[ \Rightarrow \left( \hat{i} + 2 \hat{j} + 3 \hat{k}  \right) . \left( \lambda \hat{i}  + 2 \hat{j} - 7 \hat{k}  \right) = 0\]

\[ \Rightarrow \lambda + 4 - 21 = 0\]

\[ \Rightarrow \lambda - 17 = 0\]

\[ \Rightarrow \lambda = 17\]