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प्रश्न
Find the vector equation of a plane passing through a point with position vector \[2 \hat{i} - \hat{j} + \hat{k} \] and perpendicular to the vector \[4 \hat{i} + 2 \hat{j} - 3 \hat{k} .\]
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उत्तर
\[\text{ We know that the vector equation of the plane passing through a point } \vec{a} \text{ and normal to } \vec{n} is\]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \text{ Substituting } \vec{a} = 2 \hat{ i} - \hat{j} + \hat{k} \text{ and } \vec{n} = 4 \hat{i} + 2 \hat{j} - 3 \hat{k} , \text{ we get } \]
\[ \vec{r} . \left( 4 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = \left( 2 \hat{i} - \hat{j} + \hat{k} \right) . \left( 4 \hat{i} + 2 \hat{j} - 3 \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( 4 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 8 - 2 - 3\]
\[ \Rightarrow \vec{r} . \left( 4 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 3\]
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