Let a , b , and c be three unit vectors. If their resultant vector v = a + b + c makes an identical angle with each of the vectors a , b , and c , such that = 2 3 , then the value of | a - b |^2 + | b - c |^2 + | c - a |^2 is equal to

Step-by-step solution

Given v = a + b + c . Taking the dot product of v with a : v a = | v | | a | = | v | (1) ( 2 3 ) = 2 3 | v | . Similarly, since v makes the same angle with b and c : v b = 2 3 | v | and v c = 2 3 | v | . Now, consider the dot product of v with itself: v v = v ( a + b + c ) = v a + v b + v c . | v |^2 = 2 3 | v | + 2 3 | v | + 2 3 | v | = 2 | v | . Since = 2 3 , v is not the zero vector, so we can divide by | v | to get | v | = 2 . Thus, | a + b + c |^2 = | v |^2 = 4 . Expanding | a + b + c |^2 : | a |^2 + | b |^2 +