If a , b and c are unit vectors, such that a + b + c = 0 , then 3 a b +2 b c + c a is

Step-by-step solution

a , b , c are unit vectors, then | a |=| b |=| c |=1 Given, aligned a + b + c &=0 a &=-( b + c ) aligned Squaring on both sides aligned a ² &=( b + c )² a ² &= b ² + c ² +2 b c | a |² &=| b |²+| c |²+2( b c ) ( a ² =| a |² ) aligned aligned &1=1+1+2( b c ) & b c =-1 / 2 & Similarly, a b = c a =-1 / 2 & Hence, 3 a b +2 b c + c a &=3(-1 / 2)+2(-1 / 2)+(-1 / 2) &=(3+2+1)(-1 / 2) &=6(-1 / 2)=-3 aligned