Work — definition, dot product F·d, sign convention

What is Work?

WHAT do the symbols mean?

• $F$ = magnitude of the force (N)
• $d$ = magnitude of displacement (m) — note: displacement, not distance/path length
• $\theta$ = angle between the two vectors when drawn tail-to-tail

Deriving $W = Fd\cos\theta$ from first principles

Step 1 — Resolve the force. Split $\vec F$ into components relative to $\vec d$: $F_\parallel = F\cos\theta \qquad F_\perp = F\sin\theta$ Why this step? Because energy transfer is one-dimensional along the path; decomposing isolates the part that actually moves with the object.

Step 2 — Define work as (along-motion force) × (displacement). $W = F_\parallel \, d = (F\cos\theta)\,d = Fd\cos\theta$ Why this step? The perpendicular component $F_\perp$ produces zero displacement in its direction, so it does zero work. Only $F_\parallel$ survives.

Step 3 — Recognize this is the dot product. By the very definition of the scalar product, $\vec F\cdot\vec d = Fd\cos\theta$. So work is naturally a dot product — which automatically encodes the $\cos\theta$ and the sign. $\boxed{W = \vec F\cdot\vec d = F_xd_x + F_yd_y + F_zd_z}$ Why this step? The component form lets you compute work without ever finding $\theta$ — useful when forces are given as vectors.

Sign Convention — the heart of the matter

The sign of $W$ comes entirely from $\cos\theta$:

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

• Kinetic Energy & Work-Energy Theorem — net work equals change in $\tfrac12 mv^2$
• Dot Product (Scalar Product) — the math machinery behind $Fd\cos\theta$
• Potential Energy & Conservative Forces — path-independent work
• Power — rate of doing work, $P=\vec F\cdot\vec v$
• Friction — a force that does negative work, dissipating energy as heat
• Vectors & Components — resolving $\vec F$ into parallel/perpendicular parts

Concept Map

Hinglish (regional understanding)