If a , b , c are unit vectors and the maximum value of | a - b |^2+| b - c |^2+| c - a |^2 is k , then k (2 a ^2+3 b ^2-4 c ^2 )=

Step-by-step solution

a , b and c are unit vectors. gathered | a |=| b |=| c |=1 Now, | a - b |^2+| b - c |^2+| c - a |^2 =2 ( a ^2+ b ^2+ c ^2 )-2( a b + b c + c a ) =2(1+1+1)-2( a b + b c + c a ) gathered | a - b |^2+| b - c |^2+| c - a |^2 is maximum when 2( a b + b c + c a ) is minimum using. | a + b + c |^2= a ^2+ b ^2+ c ^2+2( a b + b c + c a ) We know that, aligned & a + b + c =0, then a b + b c + c a is minimum & 0=1+1+1+2( a b + b c + c a ) & 2( a b + b c + c a )=-3 & Substitute in Eq. (i) & =6-(-3)=9 aligned Substitute in Eq.