Let a and b be two unit vectors with an angle (0, ) between them. Let E = | a - b + 2( a b )|^2 . If M is the maximum possible value of E and ₀ is the angle at which this maximum is attained, then the ordered pair (M, ₀) is equal to

Step-by-step solution

Given E = | a - b + 2( a b )|^2 . Since the cross product a b is orthogonal to both a and b , it is also orthogonal to their difference a - b . Therefore, we can expand the squared magnitude as: E = | a - b |^2 + |2( a b )|^2 E = (| a |^2 + | b |^2 - 2 a b ) + 4| a |^2| b |^2 ^2 Since a and b are unit vectors: E = (1 + 1 - 2 ) + 4 ^2 E = 2 - 2 + 4(1 - ^2 ) E = -4 ^2 - 2 + 6 Let x = . Since (0, ) , we have x (-1, 1) . The expression becomes a quadratic function in x : f(x) = -4x^2 - 2x + 6 This represents a downward