Question

Given three unit vectors a, b and c, such that a +b+c = 0, find a.b +b.c + c.a

Answer 1

a.(a +b + c ) = 0 => a.b + a.c = - a.a = - // a // ^2 = - 1 ( a is a unit vector )=> a.b + c.a = - 1 ;
where // // is Euclidean length; ( * )

b.( a + b + c ) = 0 => b.a + b.c = - b.b => a.b + b.c = - // b // ^2 = -1 ( b is a unit vector ); ( * * )

c.( a + b + c ) = 0 => c.a + c.b = -c.c => a.c + b.c = - // c // ^ 2 = - 1 ; ( c is a unit vector ) ( * * * )

From ( * ), ( * * ), ( * * * ) => 2( a. b + b.c + c.a ) = - 3 => a.b + b.c + c.a = - 3 / 2 = - 1.5