Let u and v be two unit vectors such that the scalar projection of u along the direction of u - v is 1 2 . The value of |2 u + v + u v |^2 is equal to

Step-by-step solution

Let be the angle between the unit vectors u and v . The scalar projection of u on u - v is given by u ( u - v ) | u - v | . We know that u ( u - v ) = | u |^2 - u v = 1 - . Also, | u - v | = | u |^2 + | v |^2 - 2 u v = 2 - 2 . Given that the projection is 1 2 , we have: 1 - 2 - 2 = 1 2 1 - 2(1 - ) = 1 2 1 - 2 = 1 2 Squaring both sides: 1 - 2 = 1 4 1 - = 1 2 = 1 2 Now, we need to evaluate |2 u + v + u v |^2 . Since the cross product u v is perpendicular to both u and v , it is also perpendicular to any linear combin