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Section Formula

If G denotes ...

Question

If G denotes the centroid of ∆ABC, then write the value of $\vec{G A} + \vec{G B} + \vec{G C} .$

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Solution

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices $A, B, C$ respectively. Then, the position vector of the centroid $G$ is $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$
Thus,
$\vec{G A} + \vec{G B} + \vec{G C} = \vec{a} - (\frac{\vec{a} + \vec{b} + \vec{c}}{3}) + \vec{b} - (\frac{\vec{a} + \vec{b} + \vec{c}}{3}) + \vec{c} - (\frac{\vec{a} + \vec{b} + \vec{c}}{3}) = (\vec{a} + \vec{b} + \vec{c}) - 3 (\frac{\vec{a} + \vec{b} + \vec{c}}{3}) = (\vec{a} + \vec{b} + \vec{c}) - (\vec{a} + \vec{b} + \vec{c}) = \vec{0}$

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