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  4. If ABCD be a parallelogram and M be the point of intersection of the diagonals. If O is any point, then overset→ mathopOA + overset→ mathopOB + overset→ mathopOC + overset→ mathopOD is

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Q. If $ABCD$ be a parallelogram and $M$ be the point of intersection of the diagonals. If $O$ is any point, then $ \overset{\to }{\mathop{OA}}\,+\overset{\to }{\mathop{OB}}\,+\overset{\to }{\mathop{OC}}\,+\overset{\to }{\mathop{OD}}\, $ is

KEAMKEAM 2007Vector Algebra

Solution:

We know that the diagonals of a parallelogram bisect each other. Therefore, M is the mid point of AC and BD both.
$ \therefore $ $ \overrightarrow{OA}+\overrightarrow{OC}=2\overrightarrow{OM}\text{ }and\text{ }\overrightarrow{OB}+\overrightarrow{OD}=2\overrightarrow{OM} $
$ \Rightarrow $ $ \overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=4\overrightarrow{OM} $