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