Final answer:
The question asks for the work done by a force field on a particle in three-dimensional space, but the provided examples do not match the initial query. For a constant force field, the work done is calculated by taking the dot product of the force and displacement vectors; however, in the student's question, more information is needed to solve.
Step-by-step explanation:
The question involves calculating the work done by a force field on a particle as it moves through a specific path in three-dimensional space. The force field is given by F(x, y, z) = z^2i + 4xyj + 3y^2k. To find the work done on the particle, one would typically integrate the force field along the path taken by the particle. However, the given information appears to be partly for a different question or scenario and does not provide us with enough data to solve for the work done in the scenario described at the start.
Based on examples given, if we were to consider a constant force field such as F₁ = (3 N)Î + (4 N)Ĵ and a straight-line movement from one point to another, the work done by this force could be found by taking the dot product of the force vector and the displacement vector. For a particle moving in a straight line from (0 m, 0 m) to (5 m, 6 m), the displacement vector is (5 m)Î + (6 m)Ĵ. The work done by F₁ in this case is W = F₁ · displacement = (3 N)(5 m) + (4 N)(6 m) = 15 N·m + 24 N·m = 39 joules.