3. In this question i and i are perpendicular horizontal unit vectors.] Three forces. F_(1),F_(2) and F_(3) are given by F_(1)=(5i+2j)N F_(2)=(-3i+j)N F_(3)=(ai+bj)N where a and b are constants. The forces F_(1),F_(2) and F_(3) act on a particle P of mass 4kg. Given that P rests in equilibrium on a smooth horizontal surface under the action of these three forces. (a) find the size of the angle between the direction of F_(3) and the direction of -j (4) The force F_(3) is now removed and replaced by the force F_(4) given by F_(4)=lambda (i+3j)N where lambda is a positive constant. When the three forces F_(1),F_(2) and F_(4) act on P . the acceleration of P has magnitud 3.25ms^-2 (b) Find the value of lambda

(a) To find the size of the angle between the direction of and the direction of , we can use the dot product formula for vectors. The dot product of two vectors is given by: The magnitude of is 1, so the dot product simplifies to: The magnitude of is given by: The cosine of the angle between and is given by: Therefore, the size of the angle between the direction of and the direction of is: (b) When the three forces , , and act on P, the acceleration of P has magnitude . We can use Newton's second law of motion to find the value of .According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this can be expressed as: where is the net force, is the mass of the object, and is the acceleration.In this case, the net force is the sum of the three forces , , and : Substituting the given values, we have: Combining like terms, we get: According to Newton's second law, we have: where and .Substituting the given values, we have: Simplifying the equation, we get: Comparing the coefficients of and on both sides of the equation, we have: Solving these equations, we find: Therefore, the value of is 11.