Question

Find the angle between the line \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j}  + 9 \hat{k}  \right) + \lambda\left( 2 \hat{i} + 3 \hat{j}  + 4 \hat{k}  \right)\]  and the plane  \[\vec{r} \cdot \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 5 .\]

 

Answer 1

\[ \text{ We know that the angle θ between the line } \vec{r} = \vec{a} + \lambda \vec{b} \text{ and the plane } \vec{r} . \vec{n} =\text{ dis given by} \]
\[\sin \theta = \frac{\vec{b} . \vec{n}}{\left| \vec{b} \right| \left| \vec{n} \right|} . \]
\[\text{ Here} ,\]
\[ \vec{b} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}  \text{ and }  \vec{n} = \hat{i} + \hat{j}  + \hat{k}  \]
\[ \text{ So } ,\sin \theta = \frac{\left( 2 \hat{i}  + 3 \hat{j} + 4 \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right)}{\left| 2 \hat{i} + 3 \hat{j}  + 4 \hat{k}  \right| \left| \hat{i}  + \hat{j}  + \hat{k}  \right|} = \frac{2 + 3 + 4}{\sqrt{4 + 9 + 16} \sqrt{1 + 1 + 1}} = \frac{9}{\sqrt{29} \sqrt{3}} = \frac{3 \sqrt{3}}{\sqrt{29}}\]
\[ \Rightarrow \theta = \sin^{- 1} \left( \frac{3 \sqrt{3}}{\sqrt{29}} \right)\]