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Question
Find the vector equation of the line through the origin which is perpendicular to the plane \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 3 .\]
Solution
\[\text{ The required line is perpendicular to the plane } \vec{r} . \left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right) = 3 . \]
\[\text{ Therefore, it is parallel to the normal } \hat{i} + 2 \hat{j} + 3 \hat{k} . \]
\[\text{ Thus, the required line passes through the point with position vector } \vec{a} = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} \text{ and is parallel to the vector } \vec{n} = \hat{i} -2 \hat{j} + 3 \hat{k} . \]
\[\text{ So, its vector equation is } \]
\[ r^\to = \vec{a} + \lambda \hat{n} \]
\[ \Rightarrow \vec{r} = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} + \lambda \left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)\]
\[ \Rightarrow \vec{r} = \lambda \left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)\]
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