Question

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

• A. the position of $Q$ depends on position of $P$
• B. $Q$ is a fixed point
• C. Q lies on $AB$ or $AC$
• D. None of these

Answer 1

Answer: (B)

We have
$\Rightarrow \overrightarrow{PQ}=\overrightarrow{AP}+\overrightarrow{PB}+\overrightarrow{PC}$
$\Rightarrow \overrightarrow{PQ}=\overrightarrow{AB}+\overrightarrow{PC}$
$\Rightarrow \overrightarrow{AB}=\overrightarrow{PQ}-\overrightarrow{PC}=\overrightarrow{PQ}+\overrightarrow{CP}$
$=\overrightarrow{CP}+\overrightarrow{PQ}=\overrightarrow{CQ}$
$\therefore AB=CQ$ and $AB||CQ$
$\therefore ABQC$ is a parallelogram,
$\therefore$ is a fixed point