\(\mathrm{ABC}\) is a triangle and \(\mathrm{P}\) is any point on \(\mathrm{BC}\) such that \(\overrightarrow{P Q}\) is the resultant of the vectors \(\overrightarrow{\mathrm{AP}}, \overrightarrow{\mathrm{PB}}\) and \(\overrightarrow{\mathrm{PC}}\), then

• A

  the position of \(Q\) depends on position of \(P\)
• Correct answer
  B

  \(Q\) is a fixed point
• C

  \(\mathrm{Q}\) lies on \(\mathrm{AB}\) or \(\mathrm{AC}\)
• D

  None of these

Solution

We have \(\overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{AP}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}\) \(\begin{aligned} & \Rightarrow \overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{PC}} \\ & \Rightarrow \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{PQ}}-\overrightarrow{\mathrm{PC}}=\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{CP}} \\ & =\overrightarrow{\mathrm{CP}}+\overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{CQ}} . \end{aligned}\)
\(\therefore \mathrm{AB}=\mathrm{CQ}\) and \(\mathrm{AB} \| \mathrm{CQ}\) \(\therefore \mathrm{ABQC}\) is a parallelogram. \(\therefore \mathrm{Q}\) is a fixed point.

Hence, the correct answer is option: 'B'.