Let u , v , w be three vectors representing the sides of a triangle taken in order, such that u + v + w = 0 . If | u | = 2 , | v | = 3 , and | u (2 v - w )|^2 + | u (2 v - w )|^2 = 244 , then the value of | w |^2 + | u v |^2 is equal to :

Step-by-step solution

Using Lagrange's identity, | A B |^2 + | A B |^2 = | A |^2 | B |^2 . Applying this to the given equation with A = u and B = 2 v - w : | u |^2 |2 v - w |^2 = 244 Since | u | = 2 , we have 4 |2 v - w |^2 = 244 |2 v - w |^2 = 61 From the triangle condition u + v + w = 0 , we can write w = - u - v . Substitute w into the magnitude expression: 2 v - w = 2 v - (- u - v ) = u + 3 v So, | u + 3 v |^2 = 61 Expanding the squared magnitude: | u |^2 + 9| v |^2 + 6( u v ) = 61 Substitute the given magnitudes | u | = 2 and | v |