question_answer

The three vectors \[\overset{\to }{\mathop{\text{A}}}\,\], \[\overset{\to }{\mathop{\text{B}}}\,\] and \[\overset{\to }{\mathop{\text{C}}}\,\] are represented in magnitude and direction by \[\overset{\xrightarrow[{}]{}}{\mathop{\text{OP}}}\,\], \[\overset{\xrightarrow[{}]{}}{\mathop{\text{OQ}}}\,\] and \[\overset{\xrightarrow[{}]{}}{\mathop{\text{OS}}}\,\] Fig. 2(c).52. If \[\overset{\to }{\mathop{\text{A}}}\,+\overset{\to }{\mathop{\text{B}}}\,=2\overset{\to }{\mathop{\text{C}}}\,\], show that S is the midpoint of PQ.

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.