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`.

Explanation

Let the position vector of O' with reference to O as the origin be `alpha`.
then,` O O'=alpha`
Now, `O'P_(i)=`Position vector
or `P_(i)-`Position vector of `O'=r_(i)-alpha`
`i=1,2,, . .. ,n`
Let the position vectors of `P_(1),P_(2),P_(3), . . .,P_(n)` with respect to O' as the origin be `R_(1),R_(2), . . .,R_(n)` respectively. then `R_(i)=O'`
`P_(i)=r_(i_-alpha,i=1,2, . .,n` [using eq. (i)]
Now, `a_(1)R_(1)+a_(2)R_(2)+ . .. . +a_(n)R_(n)=0`
`implies underset(i=1)overset(n)(sum)a_(i)R_(i)=0 implies underset(i=1)overset(n)(sum)a_(i)R_(i)=0 implies underset(i=1)overset(n)(sum)a_(i)(r_(i)-alpha)=0`
`implies underset(i=1)overset(n)(sum)a_(i)r_(1)-underset(i=1)overset(n)(sum)a_(i)alpha=0`
`implies 0-alpha(underset(i=1)overset(n)(sum)a_(i))=0" "[because underset(i=1)overset(n)(sum)a_(i)r_(i)=0(given)]`
`implies underset(i=1)overset(n)(sum)a_(i)=0`
thus, `underset(i=1)overset(n)(sum)a_(i)R_(i)=0` will hold good, if `underset(i=1)overset(n)(sum)a_(i)=0`.