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$G is the centroid of the triangle ABC and if G, is the centroid of another$ $triangle A_{1} B_{1} C_{1}, then value of A \vec{A_{1}} + B \vec{B_{1}} + C \vec{C_{1}} is :$

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$G {\vec{G}}_{1}$

$3 G {\vec{G}}_{1}$

None of these

answer is C.

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Detailed Solution

$Clearly A \vec{A_{1}} = A \vec{G} + G \vec{G_{1}} + G_{1} \vec{A_{1}}$

$B \vec{B_{1}} = B \vec{G} + G {\vec{G}}_{1} + G_{1} \vec{B_{1}}$

$C \vec{C_{1}} = C \vec{G} + G \vec{G_{1}} + G_{1} \vec{C_{1}}$

$Adding \underline{\underline{these}}, A \vec{A_{1}} + B \vec{B_{1}} + C \vec{C_{1}}$

$= 3 G \vec{G_{1}} + (\vec{A G} + \vec{B G} + \vec{C G}) + (G_{1} \vec{A_{1}} + G_{1} \vec{B_{1}} + G_{1} \vec{C_{1}})$

$= 3 G \vec{G_{1}} + (\vec{A G} + 2 \vec{D G}) + (G_{1} \vec{A_{1}} + 2 G_{1} \vec{D_{1}})$

$(where AD and A_{1} D_{1} are medians of the triangles ABC and A_{1} B_{1} C_{1} respectively.)$

$= 3 G \vec{G_{1}} + (\vec{A G} + \vec{G A}) + (G_{1} \vec{A_{1}} + A_{1} \vec{G_{1}})$

$= 3 G \vec{G_{1}} + \vec{0} + \vec{0} = 3 G \vec{G_{1}}$

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