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Question
If G denotes the centroid of ∆ABC, then write the value of \[\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} .\]
Solution
Let \[\overrightarrow{a} , \overrightarrow{b} , \overrightarrow{c}\] be the position vectors of the vertices A, B, C respectively.
Then, the position vector of the centroid G is \[\frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3}\]
Thus,
\[\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \overrightarrow{a} - \left( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} \right) + \overrightarrow{b} - \left( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} \right) + \overrightarrow{c} - \left( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} \right)\]
\[ = \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right) - 3 \left( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} \right)\]
\[ = \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right) - \left( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \right)\]
\[ = \overrightarrow{0}\]
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