Question

The distance of the line \[\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k}  \right)\]  from the plane \[\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5\] is

 

Options

• \[\frac{5}{3\sqrt{3}}\]

   

• \[\frac{10}{3\sqrt{3}}\]

• \[\frac{25}{3\sqrt{3}}\]

   

•  None of these

   

Answer 1

 \[\frac{10}{3\sqrt{3}}\]

\[\text{ The given line passes through the point whose position vector is } \vec{a} =2 \hat{i} -2 \hat{j}  +3 \hat{k}  .\]
\[\text{ We know that the perpendicular distance of a point P of position vector }  \vec{a} \text{ from the plane}  \vec{r} . \vec{n} =d \text{ is given by} \]
\[p = \frac{\left| \vec{a} . \vec{n} - d \right|}{\left| \vec{n} \right|}\]
\[Here, \vec{a} = 2 \hat{i} -2 \hat{j} +3 \hat{k}  ; \vec{n} = \hat{i} + 5 \hat{j}  + \hat{k}  ; d = 5\]
\[\text{ So, the required distance p is given by } \]
\[p = \frac{\left| \left( 2 \hat{i} -2 \hat{j} +3 \hat{k} \right) . \left( \hat{i} + 5 \hat{j} + \hat{k}  \right) - 5 \right|}{\left| \hat{i}  + 5 \hat{j}  + \hat{k}  \right|}\]
\[ = \frac{\left| 2 - 10 + 3 - 5 \right|}{\sqrt{1 + 25 + 1}}\]
\[ = \frac{\left| - 10 \right|}{\sqrt{27}}\]
\[ = \frac{10}{3 \sqrt{3}} \text{ units} \]