A) \[\frac{\pi }{6}\]
B) \[\frac{\pi }{4}\]
C) \[\frac{\pi }{3}\]
D) \[\frac{\pi }{2}\]
Correct Answer: D
Solution :
Direction ratios of \[OP=(0-0,\,1-0,\,\,2-0)=(0,1,2)\] Direction ratios of \[OQ=(4-0,-2-0,1-0)=(4,-2,1)\] Let \[\angle POQ=\theta ,\]then \[\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\] \[\Rightarrow \] \[\cos \theta =\frac{0-2\times 1+2\times 1}{\sqrt{0+1+4}\sqrt{16+4+1}}\] \[=\frac{0-2+2}{\sqrt{5}\sqrt{21}}=0\] \[\Rightarrow \] \[\theta =\frac{\pi }{2}\] \[\therefore \] \[\angle POQ=\frac{\pi }{2}\]
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