If a , b , c are vectors of equal magnitude such that ( a , b )= ,( b , c )= ,( c , a )= , then the minimum value of + + is

Step-by-step solution

We have, = a b | a || b | , = b c | b || c | and = c a | c || a | Also, it is given that | a |=| b |=| c |= (say), where >0 . Then, = a b ^2 ; = b c ^2 and = c a ^2 + + = 1 ^2 We know that, aligned & | a + b + c |^2=( a + b + c ) ( a + b + c ) = & [| a |^2+| b |^2+| c |^2+2( a b + b c + c a ) ] & [ ^2+ ^2+ ^2+2( a b + b c + c a ) ] 0 & [ | a + b + c |^2 0 ] & 2( a b + b c + c a ) -3 ^2 aligned Now, from Eqs. (i) and (ii), we get aligned + + & 1 ^2 (- 3 2 ^2 ) & =- 3 2 aligned Thus, the minimum value of + + is - 3 2