Question

Calculate the scalar product of the following vectors.
Prove that the sum of the vectors which connects the centre of a regular triangle with its vertices is zero.

Answer 1

Let M denote the centroid of the triangle.

The vector denoting the median from vertex $A$ is given by $\frac{1}{2}(B+C)-A$.

We know that the centroid of the triangle divides the median in the ratio $2:1$

Thus the position vector of centroid is given by $A+\frac{2}{3}(\frac{1}{2}(B+C)-A)=\frac{A+B+C}{3}=M$

$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}=A-M+B-M+C-M=A+B+C-3M=A+B+C-3.\left(\frac{A+B+C}{3}\right)=0$