Answer:
In
\[\Delta OPS\], by vectors addition,
\[\overset{\to
}{\mathop{\text{OS}}}\,\text{ }=\text{ }\overset{\to
}{\mathop{\text{OP}}}\,\text{ }+\text{ }\overset{\to }{\mathop{\text{PS}}}\,\] ??
(i)
Similarly,
in \[\Delta \text{OQS}\], we have
\[\overset{\to
}{\mathop{OS}}\,=\overset{\to }{\mathop{OQ}}\,\,+\,\overset{\to }{\mathop{PS}}\,\] ??.
(ii)
Adding
(i) and (ii), we have
\[2\overset{\to
}{\mathop{OS}}\,=\overset{\to }{\mathop{OP}}\,\,+\,\overset{\to
}{\mathop{OQ}}\,+\overset{\to }{\mathop{PS}}\,+\overset{\to }{\mathop{QS}}\,\] or
\[2\overset{\to }{\mathop{C}}\,=\overset{\to }{\mathop{A}}\,\,+\,\overset{\to
}{\mathop{B}}\,+\overset{\to }{\mathop{PS}}\,+\overset{\to }{\mathop{QS}}\,\]
or
\[(2\overset{\to }{\mathop{C}}\,-\overset{\to }{\mathop{A}}\,\,-\,\overset{\to
}{\mathop{B}}\,)=\overset{\to }{\mathop{PS}}\,+\overset{\to }{\mathop{QS}}\,\] or
\[0=\overset{\to }{\mathop{PS}}\,+\overset{\to }{\mathop{QS}}\,\] \[\left(
\because \,\,\,2\overset{\to }{\mathop{C}}\,=\overset{\to
}{\mathop{A}}\,+\overset{\to }{\mathop{B}}\, \right)\]
or
\[\overset{\to }{\mathop{PS}}\,=-\overset{\to }{\mathop{QS}}\,\]
So
S is the midpoint of PQ.
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