Let r→1,r→2,r→3,⋯,r→n be the position vectors of points P1,P2,P3,⋯,Pn relative to the origin O. If the vector equation a1r1→+a2r2→+⋯+anr→n=0 holds, then a similar equation will also hold w.r.t. to any other origin provided

a 1 = a 2 = a 3 = ⋯ = a n = 0

Let r→1,r→2,r→3,⋯,r→n be the position vectors of points P1,P2,P3,⋯,Pn relative to the origin O. If the vector equation a1r1→+a2r2→+⋯+anr→n=0 holds, then a similar equation will also hold w.r.t. to any other origin provided

a 1 = a 2 = a 3 = ⋯ = a n = 0

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Let ${\vec{r}}_{1}, {\vec{r}}_{2}, {\vec{r}}_{3}, \dots, {\vec{r}}_{n}$ be the position vectors of points $P_{1}, P_{2}, P_{3}, \dots, P_{n}$ relative to the origin O. If the vector equation $a_{1} \vec{r_{1}} + a_{2} \vec{r_{2}} + \dots + a_{n} {\vec{r}}_{n} = 0$ holds, then a similar equation will also hold w.r.t. to any other origin provided

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$a_{1} + a_{2} + \dots + a_{n} = 0$

$a_{1} + a_{2} + \dots + a_{n} = n$

$a_{1} = a_{2} = a_{3} = \dots = a_{n} = 0$

$a_{1} + a_{2} + \dots + a_{n} = 1$

answer is C.

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Detailed Solution

$Given a_{1} \vec{r_{1}} + a_{2} {\vec{r}}_{2} + \dots + a_{n} {\vec{r}}_{n} = 0$

$Now \vec{a} + \vec{r_{1}^{'}} = \vec{r_{1}} and so on$

hence,

$\begin{array}{l} a_{1} (\vec{a} + {\vec{r_{1}}}^{'}) + a_{2} (\vec{a} + \vec{r_{2}^{'}}) + \dots + a_{n} (\vec{a} + \vec{r_{n}^{'}}) = 0 \\ a_{1} \vec{r_{1}^{'}} + a_{2} {\vec{r}}_{2}^{'} + \dots + a_{n} \vec{r_{n}^{'}} + \vec{a} (a_{1} + a_{2} + \dots + a_{n}) = 0 \end{array}$