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Question
Write the vector equation of the line passing through the point (1, −2, −3) and normal to the plane \[\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right) = 5 .\]
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Solution
\[\text{ The required line is normal to the plane } \vec{r} . \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right)=5 \text{ and it is parallel to the normal vector of the plane. } \]
\[\text{ So, the required line is parallel to the vector } \vec{b} =2 \hat{i} + \hat{j} + 2 \hat{k}\]
\[\text{ It is given that the line passes through the point } (1, -2, -3) \text{ whose position vector is given by } \vec{a} = \hat{i} -2 \hat{j} -3 \hat{k} .\]
\[\text{ We know that the equation of the line passing through the point whose position vector is } \vec{a} \text{ and parallel to the vector } \vec{b} \text{ is given by } \]
\[ \vec{r} = \vec{a} + \lambda \vec{b} \]
\[ \Rightarrow \vec{r} = \left( \hat{i} - 2 \hat{j} - 3 \hat{k} \right) + \lambda \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right)\]
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