Question

If \[\overrightarrow{a}\], \[\overrightarrow{b}\], \[\overrightarrow{c}\] are the position vectors of the vertices of a triangle, then write the position vector of its centroid.

Answer 1

Let ABC be a triangle and D, E and F are the midpoints of the sides BC, CA and AB  respectively.  
Also, Let \[\overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c}\] are the position vectors of A, B, C respectively.
Then the position vectors of D, E, F are \[\left( \frac{\overrightarrow{b} + \overrightarrow{c}}{2} \right), \left( \frac{\overrightarrow{c} + \overrightarrow{a}}{2} \right), \left( \frac{\overrightarrow{a} + \overrightarrow{b}}{2} \right)\] respectively.
The position vector of a point divides AD  in the ratio of 2 ;  is \[\frac{1 . \overrightarrow{a} + 2\frac{\overrightarrow{b} + \overrightarrow{c}}{2}}{2} = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} .\]
Similarly, Position vectors of the points divides BE, CF  in the ratio of 2 : 1 are equal to \[\frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3}\].
Thus, the point dividing AD  in the ratio 2 : 1 also divides BE, CF in the same ratio.
Hence, the medians of a triangle are concurrent and the position vector of the centroid is \[\frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3}\]