If a , b , c are three vectors such that | a |=| b |=| c |= 3 and ( a + b + c )^2+( b + c - a )^2+( c + a - b )^2=36 then, |2 a -3 b +2 c |^2=

Step-by-step solution

Given that | a |=| b |=| c |= 3 and aligned & ( a + b - c )^2+( b + c - a )^2+( c + a - b )^2=36 & | a |^2+| b |^2+| c |^2+2 a b -2 b c -2 a c +| b |^2 & +| c |^2+| a |^2+2 b c -2 a b -2 c a +| c |^2+| a |^2 & +| b |^2+2 c a -2 a b -2 b c =36 & 3| a |^2+3| b |^2+3| c |^2-2( a b + b c + c a )=36 & 3[3+3+3]-2( a b + b c + c a )=36 & 27-2( a b + b c + c a )=36 aligned 2( a b + b c + c a )=-9 ... (i) Now, ( a + b + c )^2=| a |^2+| b |^2+| c |^2+2( a b + b c + c a )=3+3+9-9=0 a + b + c =0 a + c =- b ... (ii) Now, |2 a -