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Question
Write the distance of the plane \[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12\] from the origin.
Solution
\[\text{ The given equation of the plane is} \]
\[ \vec{r} . \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12 \text{ or } \vec{r} . \vec{n} = - 6, \text{ where } \vec{n} =2 \hat{i} - \hat{j} + 2 \hat{k} \]
\[\left| \vec{n} \right| = \sqrt{4 + 1 + 4} = 3\]
\[\text{ For reducing the given equation to normal form, we need to divide both sides by } \left| \vec{n} \right|. \text{ Then, we get } \]
\[ \vec{r} . \frac{\vec{n}}{\left| \vec{n} \right|} = \frac{12}{\left| \vec{n} \right|}\]
\[ \Rightarrow \vec{r} . \left( \frac{2 \hat{i} - \hat{j} + 2 \hat{k} }{3} \right) = \frac{12}{3}\]
\[ \Rightarrow \vec{r} . \left( \frac{2}{3} \hat{i} - \frac{1}{3} \hat{j} + \frac{2}{3} \hat{k} \right) = 4 . . . \left( 1 \right)\]
\[\text{ The equation of the plane in normal form is } \]
\[ \vec{r} . \hat{n} = d . . . \left( 2 \right)\]
\[( \text{ where d is the distance of the plane from the origin } )\]
\[\text{ Comparing (1) and (2) } ,\]
\[\text{ length of the perpendicular from the origin to the plane =d= 4 units } \]
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