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Question
Find the equation of the plane passing through the intersection of the planes \[\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) = 7, \vec{r} \cdot \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) = 9\] and the point (2, 1, 3).
Solution
\[\text{ The equation of the plane passing through the line of intersection of the given planes is } \]
\[ \vec{r} . \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) - 7 + \lambda \left( \vec{r} . \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) - 9 \right) = 0 \]
\[ \vec{r} . \left[ \left( 2 + 2\lambda \right) \hat{i} + \left( 1 + 5\lambda \right) \hat{j} + \left( 3 + 3\lambda \right) \hat{k} \right] - 7 - 9\lambda = 0 . . . \left( 1 \right)\]
\[\text{ This passes through } 2 \hat{i} + \hat{j} + 3 \hat{k} . \text{ So } ,\]
\[\left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) \left[ \left( 2 + 2\lambda \right) \hat{i} + \left( 1 + 5\lambda \right) \hat{j} + \left( 3 + 3\lambda \right) \hat{k} \right] - 7 - 9\lambda = 0\]
\[ \Rightarrow 4 + 4\lambda + 1 + 5\lambda + 9 + 9\lambda - 7 - 9\lambda = 0\]
\[ \Rightarrow 9\lambda + 7 = 0\]
\[ \Rightarrow \lambda = \frac{- 7}{9}\]
\[\text{ Substituting this in (1), we get } \]
\[ \vec{r} . \left[ \left( 2 + 2 \left( \frac{- 7}{9} \right) \right) \hat{i} + \left( 1 + 5 \left( \frac{- 7}{9} \right) \right) \hat{j} + \left( 3 + 3 \left( \frac{- 7}{9} \right) \right) \hat{k} \right] - 7 - 9 \left( \frac{- 7}{9} \right) = 0\]
\[ \Rightarrow \vec{r} . \left( 4 \hat{i} - 26 \hat{j} + 6 \hat{k} \right) = 0\]
\[ \Rightarrow \vec{r} . \left( 2 \hat{i} - 13 \hat{j} + 3 \hat{k} \right) = 0\]
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