If G and G' are the centroids of the triangle | vedclass

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(D) Let the position vectors of the vertices be $\vec{a}, \vec{b}, \vec{c}$ and $\vec{a'}, \vec{b'}, \vec{c'}$.The centroids $G$ and $G'$ are given by

Vedclass · Accepted Answer

$3\overrightarrow{GG'}$

Vedclass · Answer

(D) Let the position vectors of the vertices be $\vec{a}, \vec{b}, \vec{c}$ and $\vec{a'}, \vec{b'}, \vec{c'}$.The centroids $G$ and $G'$ are given by $\vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$ and $\vec{g'} = \frac{\vec{a'} + \vec{b'} + \vec{c'}}{3}$.Therefore,$\vec{a} + \vec{b} + \vec{c} = 3\vec{g}$ and $\vec{a'} + \vec{b'} + \vec{c'} = 3\vec{g'}$.We need to find $\overrightarrow{AA'} + \overrightarrow{BB'} + \overrightarrow{CC'} = (\vec{a'} - \vec{a}) + (\vec{b'} - \vec{b}) + (\vec{c'} - \vec{c})$.Rearranging the terms,we get $(\vec{a'} + \vec{b'} + \vec{c'}) - (\vec{a} + \vec{b} + \vec{c}) = 3\vec{g'} - 3\vec{g} = 3(\vec{g'} - \vec{g})$.Since $\vec{g'} - \vec{g} = \overrightarrow{GG'}$,the result is $3\overrightarrow{GG'}$.

If $G$ and $G'$ are the centroids of the triangles $ABC$ and $A'B'C'$ respectively,then $\overrightarrow{AA'} + \overrightarrow{BB'} + \overrightarrow{CC'} = $

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