A) \[{{\sin }^{-1}}\left( \frac{2}{\sqrt{3}} \right)\]
B) \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]
C) \[\frac{\pi }{4}\]
D) \[{{\sin }^{-1}}\left( \frac{1}{3\sqrt{3}} \right)\]
E) \[{{\sin }^{-1}}(2)\]
Correct Answer: D
Solution :
If\[\theta \]is the angle between the line \[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\] and the plane\[ax+by+cz+d=0,\]then \[\sin \theta =\frac{al+mb+cn}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\sqrt{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}}\] Given line \[\frac{x-3}{2}=\frac{y-1}{1}=\frac{x+4}{-2},\] Plane \[x+y+z+5=0\] \[\therefore \]\[\sin \theta =\frac{2+1-2}{\sqrt{4+1+4}\sqrt{1+1+1}}=\frac{1}{3\sqrt{3}}\] \[\Rightarrow \] \[\theta ={{\sin }^{-1}}\left( \frac{1}{3\sqrt{3}} \right)\]
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