Work from Force and Displacement Components

Work is the dot product of force and displacement, so in components it is simply F_xd_x + F_yd_y — no angle to measure and no cosine to look up. Each axis contributes independently, and the contributions can cancel: a force of (12, −5) N acting through a displacement of (3, 4) m does 36 − 20 = 16 J of work, the downward pull eating into what the forward push delivered. A component pair that sums to zero means the force was perpendicular to the motion and did no work at all, which is why a satellite in a circular orbit neither gains nor loses energy despite being pulled on the whole way around.

Gaspard-Gustave de Coriolis named the quantity travail in 1829 and fixed the modern convention of force times distance; the joule was attached to it later, honouring James Prescott Joule's paddle-wheel measurements of the mechanical equivalent of heat. The component form makes the sign convention obvious in a way the |F||d| cos θ form does not: negative work simply means the force opposed the motion, exactly what friction and braking do.