Let u and v be two unit vectors with an acute angle between them. A vector w is defined as w = 4 u + 5 v + 2 ( u v ) . If 2( w u ) - ( w v ) = 4 , then the value of 35 ^2 is

Step-by-step solution

Given w = 4 u + 5 v + 2 ( u v ) . Taking the dot product of w with u : w u = 4( u u ) + 5( v u ) + 2 (( u v ) u ) Since u is a unit vector, u u = 1 . The cross product u v is orthogonal to u , so ( u v ) u = 0 . Thus, w u = 4 + 5 . Similarly, taking the dot product of w with v : w v = 4( u v ) + 5( v v ) + 2 (( u v ) v ) w v = 4 + 5 . Substitute these into the given condition 2( w u ) - ( w v ) = 4 : 2(4 + 5 ) - (4 + 5) = 4 8 + 10 - 4 - 5 = 4 3 + 6 = 4 6 = 1 = 1 6 . We need to find 35 ^2 . Using the identity ^2 = ^