If a , b , c are three vectors such that |a| =3, |b| =5, |c| =7 then | a - b |^2+| b - c |^2+| c - a |^2 does not exceed

Step-by-step solution

Given vectors a , b , and c with magnitudes | a | = 3 , | b | = 5 , and | c | = 7 , we compute E = | a - b |^2 + | b - c |^2 + | c - a |^2 . Using the identity | x - y |^2 = | x |^2 + | y |^2 - 2 x y , we expand: E = (| a |^2 + | b |^2 - 2 a b ) + (| b |^2 + | c |^2 - 2 b c ) + (| c |^2 + | a |^2 - 2 c a ) Summing terms yields E = 2(| a |^2 + | b |^2 + | c |^2) - 2( a b + b c + c a ) . With | a |^2 + | b |^2 + | c |^2 = 9 + 25 + 49 = 83 , we have E = 166 - 2( a b + b c + c a ) . From | a + b + c |^2 = | a |^2 + | b