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प्रश्न
The line \[\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right)\] is parallel to the plane \[\vec{r} \cdot \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 4 .\] Find m.
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उत्तर
\[\text{ The given line is parallel to the vector } \vec{b} = 2 \hat{i} - m \hat{j} - 3 \hat{k} \text{ and the given plane is normal to the vector } \vec{n} = m \hat{i} + 3 \hat{j} + \hat{k} . \]
\[\text{ If the line is parallel to the plane, the normal to the plane is perpendicular to the line.} \]
\[ \Rightarrow \vec{b} \perp \vec{n} \]
\[ \Rightarrow \vec{b} . \vec{n} = 0\]
\[ \Rightarrow \left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right) . \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 0\]
\[ \Rightarrow 2m - 3m - 3 = 0\]
\[ \Rightarrow - m - 3 = 0\]
\[ \Rightarrow m = - 3\]
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