Calculate the scalar product of the following vectors.
M is the point of intersection of the medians of a triangle ABC. Prove that $- - \to M A + - - \to M B + - - \to M C = 0.$

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Solution

M is given to be 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 $A + \frac{2}{3} (\frac{1}{2} (B + C) - A) = \frac{A + B + C}{3} = M$

$\to M A + \to M B + \to M C = A - M + B - M + C - M = A + B + C - 3 M = A + B + C - 3. (\frac{A + B + C}{3}) = 0$

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