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प्रश्न
Determine the value of λ for which the following planes are perpendicular to each other.
उत्तर
` \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.`
\[\text{ Here} , \vec{n_1} = \hat{i}| + 2 \hat{j} + 3 \hat{k} ; \vec{n_2} = \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \]
\[\text{ The given planes are perpendicular.} \]
\[ \Rightarrow \vec{n_1} . \vec{n_2} = 0\]
\[ \Rightarrow \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) . \left( \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \right) = 0\]
\[ \Rightarrow \lambda + 4 - 21 = 0\]
\[ \Rightarrow \lambda - 17 = 0\]
\[ \Rightarrow \lambda = 17\]
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