LinAlg4.1.2

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If G is the centroid of the triangle ABC,prove that \bar{GA}+\bar{GB}+\bar{GC}=\bar{O}\, where A,B,C\, are the vertices of the triangle ABC and \bar{O}\, is the point vector

Let A=\bar{a},B=\bar{b},C=\bar{c}\, be the vertices of the given triangle.

Then the position vector of the centroid G=\frac{1}{3}(\bar{a}+\bar{b}+\bar{c})\,

\bar{GA}=\bar{A}-\bar{G}=\bar{a}-[\frac{1}{3}(\bar{a}+\bar{b}+\bar{c})]=\frac{1}{3}(2\bar{a}-\bar{b}-\bar{c})\,

Similarly \bar{GB}=\frac{1}{3}(2\bar{b}-\bar{a}-\bar{c}),\bar{GC}=\frac{1}{3}(2\bar{c}-\bar{a}-\bar{b})\,

\bar{GA}+\bar{GB}+\bar{GC}=\frac{1}{3}(2\bar{a}-\bar{b}-\bar{c})+\frac{1}{3}(2\bar{b}-\bar{a}-\bar{c})+\frac{1}{3}(2\bar{c}-\bar{a}-\bar{b})=\frac{1}{3}(\bar{O})=\bar{O}\,


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