If a , b and c are any three non-zero vectors, then ( a +2 b + c ) [( a - b ) ( a - b - c )]=

Step-by-step solution

aligned & ( a +2 b + c ) [( a - b ) ( a - b - c )] & =( a +2 b + c ) ( a a - a b - a c - b a + b b + b c ) & =( a +2 b + c ) ( 0 - a b - a c + a b + 0 + b c ) & =( a +2 b + c ) ( c a + b c ) & = a ( c a )+ a ( b c )+2 b ( c a )+2 b ( b c ) & + c ( c a )+ c ( b c ) & = 0 + a ( b c )+2 b ( c a )+2 0-0-0 & = [ array lll a & b & c array ]+2 [ array lll b & c & a array ] & = [ array lll a & b & c array ]+2 [ array lll a & b & c array ] & =3 [ array lll a & b & c array ] & aligned