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Let r(1),r(2),r(3), . . .,r(n) be the po...

Let `r_(1),r_(2),r_(3), . . .,r_(n)` be the position vectors of points `P_(1),P_(2),P_(3), . . .,P_(n)` relative to an origin O. show that if then a similar equation will also hold good with respect to any other origin O'. If `a_(1)+a_(2)+a_(3)+ . . .+a_(n)=0`.

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Let A,B and C be the three points whose position vectors referred to O are `r_(1),r_(2)` and `r_(3)` respectively.
`AB=OB-OA=r_(2)r_(1)`.
`BC=OC-OB=r_(3)-r_(2)`
Now, if A,B and C are collinear points, then AB and AC are the same line and BC=`lamda(AC)`
`implies(r_(3)-r_(2))=lamda(r_(2)-r_(1))`
`implies r_(3)=-lamdar_(1)+(lamda+1)r_(2)`
`therefore r_(3)=-lamdar_(1)+mr_(2)`
where, `l=-lamda and m=lamda+1`
`implies l+m=-lamda+(lamda+1)=1`
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