Let ` vec r_1, vec r_2, vec r_3, , vec r_n` be the position vectors of points `P_1,P_2, P_3 ,P_n` relative to the origin `Odot` If the vector equation `a_1 vec r_1+a_2 vec r_2++a_n vec r_n=0` hold, then a similar equation will also hold w.r.t. to any other origin provided a. `a_1+a_2+dot+a_n=n` b. `a_1+a_2+dot+a_n=1` c. `a_1+a_2+dot+a_n=0` d. `a_1=a_2=a_3dot+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`