Question

Write the distance of the plane  \[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12\] from the origin.

  

Answer 1

\[\text{ The given equation of the plane is} \]

\[ \vec{r} . \left( 2 \hat{i}  - \hat{j} + 2 \hat{k} \right) = 12 \text{ or } \vec{r} . \vec{n} = - 6, \text{ where } \vec{n} =2 \hat{i}  - \hat{j}  + 2 \hat{k}  \]

\[\left| \vec{n} \right| = \sqrt{4 + 1 + 4} = 3\]

\[\text{ For reducing the given equation to normal form, we need to divide both sides by } \left| \vec{n} \right|. \text{ Then, we get } \]

\[ \vec{r} . \frac{\vec{n}}{\left| \vec{n} \right|} = \frac{12}{\left| \vec{n} \right|}\]

\[ \Rightarrow \vec{r} . \left( \frac{2 \hat{i}  - \hat{j}  + 2 \hat{k} }{3} \right) = \frac{12}{3}\]

\[ \Rightarrow \vec{r} . \left( \frac{2}{3} \hat{i}  - \frac{1}{3} \hat{j} + \frac{2}{3} \hat{k}  \right) = 4 . . . \left( 1 \right)\]

\[\text{ The equation of the plane in normal form is } \]

\[ \vec{r} . \hat{n}  = d . . . \left( 2 \right)\]

\[( \text{ where d is the distance of the plane from the origin } )\]

\[\text{ Comparing (1) and (2) } ,\]

\[\text{ length of the perpendicular from the origin to the plane =d= 4 units } \]