If a , b and c are vectors with magnitudes 2,3 and 4 respectively, then the best upper bound of | a - b |^2+| b - c |^2+| c - a |^2 among the given values is

Step-by-step solution

Given, | a |=2,| ~b |=3 and | c |=4 aligned & aligned & |a-b|^2+|b-c|^2+|c-a|^2 &= a^2+b^2-2 a b+b^2+c^2-2 b c & +c^2+a^2-2 a c &= 2 (a^2+b^2+c^2 )-2(a b+b c+c a) & array rl 2 & 2(a b+b c+c a) array &=2 (a^2+b^2+c^2 ) & - (|a-b|^2+|b-c|^2+|c-a|^2 ) aligned aligned We know, (a+b+c)^2 0 aligned & a^2+b^2+c^2+2(a b+b c+c a) 0 & a^2+b^2+c^2+2 (a^2+b^2+c^2 ) & - (|a-b|^2+|b-c|^2+|c-a|^2 ) 0 & | a - b |^2+| b - c |^2+| c - a |^2 & 3 (a^2+b^2+c^2 ) & |a-b|^2+|b-c|^2+|c-a|^2 & 3 (2^2+3^2+4^2 ) & |a-b|^2+|b-c|^2+|c-a|^2 & 3