Question

State when the line \[\vec{r} = \vec{a} + \lambda \vec{b}\]  is parallel to the plane  \[\vec{r} \cdot \vec{n} = d .\]Show that the line  \[\vec{r} = \hat{i}  + \hat{j}  + \lambda\left( 3 \hat{i}  - \hat{j}  + 2 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .\]   Also, find the distance between the line and the plane.

 

 

Answer 1

\[\text{ The given plane passes through the point with position vector } \vec{a} = \hat{i}  + \hat{j}  + 0 \hat{k}  \text{ and is parallel to the vector }  \vec{b} = 3 \hat{i}  - \hat{j}  + 2 \hat{k}  . \]
\[\text{ The given plane is }  \vec{r} .\left( 2 \hat{j}  + \hat{k}  \right)=3 \text{ or }  \vec{r} . \vec{n} =d\]
\[\text{ So, normal vector} , \vec{n} =0 \hat{i}  + 2 \hat{j} + \hat{k}  \text{ and } d = 3\]
\[\text{ Now } , \vec{b} . \vec{n} = \left( 3 \hat{i}  - \hat{j}  + 2 \hat{k}  \right) . \left( 0 \hat{i}  + 2 \hat{j}  + \hat{k}  \right) = 0 - 2 + 2 = 0\]
\[\text{ So } , \vec{b} \text{ is perpendicular to } \vec{n} .\]
\[\text{ Hence,the given line is parallel to the given plane } .\]
\[\text{ The distance between the line and the parallel plane is the distance between any point on the line and the given plane. The plane passes through the point } \vec{a} = \hat{i} + \hat{j}  + 0 \hat{k} . \]
\[\text{ The perpendicular distance from the given line to the plane is } \]
\[d = \frac{\left| \vec{a} . \vec{n} - d \right|}{\left| \vec{n} \right|}\]
\[ = \frac{\left| \left( \hat{i}  + \hat{j}  + 0 \hat{k} \right) . \left( 0 \hat{i}  + 2 \hat{j}  + \hat{k}  \right) - 3 \right|}{\left| 0 \hat{i}  + 2 \hat{j}  + \hat{k}  \right|}\]
\[ = \frac{\left| 0 + 2 + 0 - 3 \right|}{\sqrt{0 + 2^2 + 1^2}}\]
\[ = \frac{1}{\sqrt{5}} \text{ units } \]