Question

Show that the line whose vector equation is \[\vec{r} = 2 \hat{i}  + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i}  + 3 \hat{j}  + 4 \hat{k}  \right)\] is parallel to the plane whose vector  \[\vec{r} \cdot \left( \hat{i} + \hat{j}  - \hat{k}  \right) = 7 .\]  Also, find the distance between them.

  

Answer 1

\[ \text{ The given plane passes through the point with position vector } \vec{a} = 2 \hat{i}  + 5 \hat{j} + 7 \hat{k}  \text { and is parallel to the vector }  \vec{b} = \hat{i} + 3 \hat{j}  + 4 \hat{k} .\]
\[ \text{ The given plane is } \vec{r} .\left( \hat{i}  + \hat{j}  - \hat{k}  \right)= 7 \text{ or .cc } \]
\[\text{ So, the normal vector } , \vec{n} = \hat{i}  + \hat{j}  - \hat{k}  \text{ and } d = 7 . \]
\[\text{ Now } , \vec{b} . \vec{n} = \left( \hat{i}  + 3 \hat{j}  + 4 \hat{k} \right) . \left( \hat{i}  + \hat{j} - \hat{k}  \right) = 1 + 3 - 4 = 4 - 4 = 0\]
\[\text{So} , \vec{b} \text{ is perpendicular to }  \vec{n} .\]
\[\text{ So, the given line is parallel to the given plane } .\]
\[ \text{ The distance between the line and the parallel plane . Then } ,\]
\[d = \text{ length of the perpendicular from the point } \vec{a} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k}  \text{ to the plane } \vec{r} . \vec{n} =d\]
\[d = \frac{\left| \vec{a} . \vec{n} - d \right|}{\left| \vec{n} \right|}\]
\[ = \frac{\left| \left( 2 \hat{i}  + 5 \hat{j}  + 7 \hat{k}  \right) . \left( \hat{i}  + \hat{j}  - \hat{k} \right) - 7 \right|}{\left| \hat{i}  + \hat{j}  - \hat{k}  \right|}\]
\[ = \frac{\left| 2 + 5 - 7 - 7 \right|}{\sqrt{1 + 1 + 1}}\]
\[ = \frac{7}{\sqrt{3}} units\]