Find the angle between the line \vec{r}= \left ( 2\hat{i}-\hat{j}+3\hat{k} \right )+\lambda \left (3\hat{i}-\hat{j}+2\hat{k} \right )  and the palne \vec{r}\cdot \left ( \hat{i}+\hat{j}+\hat{k} \right )= 3

 

 

 

 
 
 
 
 

Answers (1)

\vec{r}= \left ( 2\hat{i}-\hat{j}+3\hat{k} \right )+\lambda \left ( 3\hat{i}-\hat{j}+2\hat{k} \right )
\vec{r}\cdot \left ( \hat{i}+\hat{j}+\hat{k} \right )= 3
The required angle between the line and plane is
 \sin \theta = \frac{\left | \vec{m}\cdot \vec{b} \right |}{\left | \vec{m} \right |\left | \vec{b} \right |}
ie\; \theta = \sin^{-1}\frac{\left | \left ( \hat{i}+\hat{j}+\hat{k}\right ) \cdot \left ( 3\hat{i}-\hat{j}+2\hat{k} \right )\right |}{\left | \hat{i}+\hat{j} +\hat{k} \right |\left | 3\hat{i}-\hat{j}+2\hat{k} \right |}
\theta = \sin^{-1}\frac{\left | 3-1+2 \right |}{\sqrt{1+1+1}\sqrt{9+1+4}}
\theta = \sin^{-1}\left ( \frac{4}{\sqrt{42}} \right )
 

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