Question

For $O$ being the origin and $3$ points $P,Q$ and $R$ lie on a plane. If $\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$, then $P,Q,R$ are

• A. the vertices of an equilateral triangle
• B. the vertices of an isoceles triangle
• C. collinear
• D. none of these

Answer 1

The correct option is C collinear

$\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$
$\Rightarrow -\overrightarrow{OP}+\overrightarrow{OQ}=-\overrightarrow{OQ}+\overrightarrow{OR}$
$\Rightarrow \overrightarrow{OQ}=\frac{\overrightarrow{OP}+\overrightarrow{OR}}{2}$
Clearly $Q$ is the mid point of $P$ and $R$
Hence $P,Q,R$ are collinear.