A) \[(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC})sin2A\]
B) \[3\,\overrightarrow{OG}\], where G is the centroid of triangle ABC
C) \[\overrightarrow{0}\]
D) none of these
Correct Answer: C
Solution :
[c] The position vector of the point O with respect to itself is \[\frac{\overrightarrow{OA}\sin 2A+\overrightarrow{OB}\sin 2B+\overrightarrow{OC}\sin 2C}{\sin 2A+\sin 2B+\sin 2C}\] \[\Rightarrow \frac{\overrightarrow{OA}\sin 2A+\overrightarrow{OB}\sin 2B+\overrightarrow{OC}\sin 2C}{\sin 2A+\sin 2B+\sin 2C}=\vec{0}\]Or \[\overrightarrow{OA}\sin 2A+\overrightarrow{OB}\sin 2B+\overrightarrow{OC}\sin 2C=\vec{0}\]
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