Let vecr_1, vecr_2,……vecr_n` be the position of points `P_1,P_2,………,P_n` respectively relative to an origin O. Show that if the vector equation `a_1vecr_1+a_2vecr_2+..+a_nvecr_n=vec0` holds, then a similar equation will also hold good wilth respect to any other origin if `a_1+a_2+......+a_n=0`

Options

A. `a_1+ a_2 + ….+ a_n=n`
B. `a_1 + a_2+ ….+ a_n=1`
C. `a_1 + a_2+… + a_n=0`
D. `a_1 = a_2 = a_3= …. = a_n=0`

Answer

C

Explanation

Given `a_1vecr_1 + a_2vecr_2+…+ a_nvecr_n=0`

Now `veca + vecr'_1= vecr'_1` and so on
Hence,
`a_1 (veca+ vecr'_1)+ a_2 (veca + vecr'_2) + … + a_n (veca+ vecr'_n)=0`
`a_1vecr'_1+ a_2 vecr'_2 +…. + a_n vecr'_n + veca(a_1 + a_2 + .... + a_n) =0`
Hence, `a_1vecr'_1+ a_2 vecr'_2 +....+ a_nvecr'_n =0`
if `a_1 + a_2 +...+ a_n =0`