= Work done by a constant force
{title2=$W=Fd\cos\theta$}
For a fixed <force> $\mathbf F$ and endpoint <displacement> $\Delta\mathbf r$, the <line integral> reduces to a <dot product>:
$$
W=\mathbf F\cdot\Delta\mathbf r=Fd\cos\theta,
\qquad F=|\mathbf F|,\quad d=|\Delta\mathbf r|.
$$
Here $\theta$ is the <angle> between the <force> and the <displacement>. For straight motion, $d$ is also the distance traveled. A $3\,\mathrm N$ <force> acting along a $4\,\mathrm m$ <displacement> does $12\,\mathrm J$ of <work>. Turning the same <force> perpendicular to the <displacement> gives $\cos(\pi/2)=0$ and zero <work>; reversing it gives $\cos\pi=-1$ and $-12\,\mathrm J$. At $\theta=\pi/3$, the <work> is $6\,\mathrm J$. These signs describe <energy> transferred to or from the moving body. https://openstax.org/books/university-physics-volume-1/pages/7-1-work[OpenStax University Physics 1, §7.1: Constant-force work]{ref}
\Image[/work-force-and-displacement.png]
{title=<Work> from parallel, perpendicular, and opposing <forces>}
{description=Each left panel shows a $3\,\mathrm N$ <force> and a rightward $4\,\mathrm m$ <displacement>. Each right panel plots the along-path <force> component against distance; its signed area is the <work>: $12$, $0$, or $-12\,\mathrm J$. The arrows compare directions; their lengths do not compare quantities with different units. The three panels prescribe motion to illustrate <work>; they are separate cases, not successive stages of one journey.}
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