NIMCET 2010 — Mathematics PYQ

Let the position vectors of points $A$, $B$, and $C$ with respect to the origin $P$ be $\overrightarrow{PA}$, $\overrightarrow{PB}$, and $\overrightarrow{PC}$ respectively.

Step 2: Apply the Midpoint Formula

Since $C$ is the midpoint of the line segment $AB$, the position vector of $C$ (with respect to $P$) is the average of the position vectors of $A$ and $B$.

The formula for the midpoint $C$ is:

Step 3: Simplify the Equation

To find the relationship asked in the options, we multiply both sides by $2$:

Rearranging the equation:

Geometrical Interpretation:

In $△PAB$, if $C$ is the midpoint of the base $AB$, then $PC$ is the median. The sum of the vectors forming the sides ($\overrightarrow{PA}$ and $\overrightarrow{PB}$) is equal to twice the median vector ($\overrightarrow{PC}$).

Correct Option:

(b) $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PC}$