Let ABCDEF be a regular hexagon. If AB = a and BC = b , and the vectors v ₁ = AC + CE and v ₂ = BD - AB are perpendicular to each other, then the sum of all possible values of is :

Step-by-step solution

In the regular hexagon ABCDEF , let the side length be a . Thus, | a | = | b | = a . The angle between the vectors AB and BC when placed tail-to-tail is 60^ . Therefore, a b = a a 60^ = a^2 2 . We can express the required diagonals in terms of a and b : AC = AB + BC = a + b CD = b - a CE = CD + DE = ( b - a ) - a = b - 2 a BD = BC + CD = b + ( b - a ) = 2 b - a Now, substitute these into v ₁ and v ₂ : v ₁ = ( a + b ) + ( b - 2 a ) = ( - 2) a + ( + 1) b v ₂ = (2 b - a ) - a = -( + 1) a + 2 b Since v ₁ and v ₂ are pe