Let u and v be two vectors such that | u | = 2 . If the value of the expression |(2 u + v ) ( u - 2 v )|^2 + 25( u v )^2 is equal to 900 , then the value of | v |^2 is equal to ______

Step-by-step solution

First, expand the cross product term using the distributive property and the anti-commutative property of the cross product ( a b = - b a ): (2 u + v ) ( u - 2 v ) = 2( u u ) - 4( u v ) + ( v u ) - 2( v v ) Since the cross product of a vector with itself is zero, this simplifies to: = -4( u v ) - ( u v ) = -5( u v ) Now, square the magnitude of this result: |-5( u v )|^2 = 25| u v |^2 Substitute this back into the given expression: 25| u v |^2 + 25( u v )^2 = 900 Factor out 25 and apply Lagrange's identity, | u v |