If $C$ is the mid point of $A B$ and $P$ is any point outside $A B$, then

If $C$ is the mid point of $A B$ and $P$ is any point outside $A B$, then
  1. $\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}=2 \overrightarrow{\mathrm{PC}}$
  2. $\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+2 \overrightarrow{\mathrm{PC}}=0$
  3. $\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+ \overrightarrow{\mathrm{PC}}=0$
  4. None of these

Solution


$ \begin{aligned} & \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{CP}}=0 \\ & \overline{\mathrm{PB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CP}}=0 \end{aligned} $ Adding, we get $ \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{BC}}+2 \overrightarrow{\mathrm{CP}}=0 $ Since $\overrightarrow{A C}=-\overrightarrow{B C}$ $ \begin{aligned} & \& \overrightarrow{\mathrm{CP}}=-\overrightarrow{\mathrm{PC}} \\ & \Rightarrow \overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}-2 \overrightarrow{\mathrm{PC}}=0 . \end{aligned} $

Asked in: JEE Main 2005

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