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Question
Write the plane \[\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 14\] in normal form.
Solution
\[\text{ The given equation of the plane is } \]
\[ \vec{r} . \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 14 \text{ or } \vec{r} . \vec{n} = 14, \text{ where } \vec{n} =2 \hat{i} + 3 \hat{j} - 6 \hat{k} \]
\[\left| \vec{n} \right| = \sqrt{4 + 9 + 36} = 7\]
\[F\text{ or reducing the given equation to normal form, we need to divide it by } \left| \vec{n} \right|. \text{ Then, we get } \]
\[ \vec{r} . \frac{\vec{n}}{\left| \vec{n} \right|} = \frac{14}{\left| \vec{n} \right|}\]
\[ \Rightarrow \vec{r} . \left( \frac{2 \hat{i} + 3 \hat{j} - 6 \hat{k} }{7} \right) = \frac{14}{7}\]
\[ \Rightarrow \vec{r} . \left( \frac{2}{7} \hat{i} + \frac{3}{7} \hat{j} - \frac{6}{7} \hat{k} \right) = 2, \text{ which is the required normal form} .\]
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