A) \[{{\sin }^{-1}}\left( \frac{15}{21} \right)\]
B) \[{{\cos }^{-1}}\left( \frac{16}{21} \right)\]
C) \[{{\sin }^{-1}}\left( \frac{16}{21} \right)\]
D) \[\frac{\pi }{2}\]
E) \[{{\cos }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]
Correct Answer: C
Solution :
The given line is \[\overrightarrow{r}=(1+2\mu )\hat{i}+(2+\mu )\hat{j}+(2\mu -1)\hat{k}\] \[=(\hat{i}+2\hat{j}-\hat{k})+\mu (2\hat{i}+\hat{j}+2\hat{k})\] \[\therefore \]Equation of line in cartesian form is \[\frac{x-1}{2}=\frac{y-2}{1}=\frac{z+1}{2}\] \[\therefore \]Angle between line and a plane is \[\sin \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\] \[=\frac{2\times 3+1\times (-2)+2\times 6}{\sqrt{4+1+4}\sqrt{9+4+36}}\] \[=\frac{6-2+12}{21}=\frac{16}{21}\] \[\Rightarrow \] \[\theta ={{\sin }^{-1}}\left( \frac{16}{21} \right)\]
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