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The Centroid

If G and G' b...

Question

If $G$ and $G^{'}$ be the centroids of the triangles $A B C$ and $A^{'} B^{'} C^{'}$ respectively, then $- - \to A A^{'} + - - \to B B^{'} + - - \to C C^{'} =$

A

3 - - \to G^{'} G

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B

3 - - \to G G^{'}

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C

2 - - \to G G^{'}

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D

\frac{3}{2} - - \to G G^{'}

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Solution

The correct option is B $3 - - \to G G^{'}$
Let position vector of $A (\to a), B (\to b), C (\to c), A^{'} (^{'}), B^{'} (^{'})$ and $C^{'} (^{'})$
Now position vector of $G = \frac{\to a + \to b + \to c}{3}$
Position vector of $G^{'} = \frac{^{'} +^{'} +^{'}}{3}$
Now $- - \to A A^{'} =^{'} - \to a, - - \to B B^{'} =^{'} - \to b, - - \to C C^{'} =^{'} - \to c$
So $- - \to A A^{'} + - - \to B B^{'} + - - \to C C^{'} = (^{'} +^{'} +^{'}) - (\to a + \to b + \to c)$
$\Rightarrow - - \to A A^{'} + - - \to B B^{'} + - - \to C C^{'} = 3 ⎛ ⎝ - \frac{\to a + \to b + \to c}{3} + \frac{^{'} +^{'} +^{'}}{3} ⎞ ⎠$
$\Rightarrow - - \to A A^{'} + - - \to B B^{'} + - - \to C C^{'} = 3 - - \to G G^{'}$

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