A force F acts on a particle moving a displacement d at angle φ to the displacement direction. What is the work done by the force?

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Work is the transfer of energy caused by a force acting along the motion. For a constant force, the amount of work depends on how much of the force points in the same direction as the displacement. If φ is the angle between the force and the displacement, the component of the force along the path is F cos φ. Multiply that by the displacement d to get the work: W = F d cos φ. This also fits intuitive cases: if the force is fully along the motion (φ = 0), you get W = F d; if the force is perpendicular (φ = 90°), no work is done; if it opposes the motion (φ > 90°), the work is negative. The other expressions become correct only in special or different setups (for example, F d would require φ = 0, and F d cos(90° − φ) equals F d sin φ, which is not the general relation).