Mathematics

Mathematics

Problem

Let r1,r2,r3,…,rnr_1, r_2, r_3, \dots, r_n be the position vectors of points P1,P2,P3,…,PnP_1, P_2, P_3, \dots, P_n relative to an origin OO . Show that if the vectors equation a1r1+a2r2+⋯+anrn=0a_1 r_1 + a_2 r_2 + \dots + a_n r_n = 0 holds, then a similar equation will also hold good with respect to any other origin O′O' , if a1+a2+a3+⋯+an=0a_1 + a_2 + a_3 + \dots + a_n = 0 .
Intuitive explanation

This problem asks you to prove that a linear dependence among position vectors is origin-independent under a special condition. The key insight is that when you shift the origin, every position vector gets shifted by the same amount; if the coefficients sum to zero, that common shift cancels out perfectly. Think of it as a balance condition—when weights sum to zero, the center of mass doesn't move, so the relationship holds no matter where you measure from.

Intuitive explanation

This problem asks you to prove that a linear dependence among position vectors is origin-independent under a special condition. The key insight is that when you shift the origin, every position vector gets shifted by the same amount; if the coefficients sum to zero, that common shift cancels out perfectly. Think of it as a balance condition—when weights sum to zero, the center of mass doesn't move, so the relationship holds no matter where you measure from.

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