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

This is the D1 Foundations child of the Work topic. The parent note throws around arrows, angles, $\cos \theta$, dot products, and $F_x{d}_x+F_y{d}_y$. Here we build each one from nothing, in an order where every piece leans on the piece before it. A reader who has never seen an arrow-with-a-hat should finish this page ready.

0. Number vs. arrow — scalar and vector

Before any physics, one split matters.

Look at the figure. The pale-yellow arrow has a tail (where it starts) and a head (the pointy end). Its length is the yellow bar labelled "magnitude"; its tilt is the direction. Two arrows are the same vector if they have the same length and the same tilt — even if drawn in different places.

We write a vector with a little arrow on top: $\vec{F}$, $\vec{d}$. We write its size only (a scalar) with the same letter, no arrow: $F$, $d$. So $F$ is always $\ge 0$; it is just the length of $\vec{F}$.

1. Displacement $\vec{d}$ — the arrow from start to finish

The figure shows a wandering chalk path (the actual route walked) versus the straight blue arrow $\vec{d}$ (start-to-finish). These are different:

• Path length / distance = how far you actually travelled along the squiggle (always a scalar, always grows).
• Displacement $\vec{d}$ = the shortcut arrow (a vector, can even be zero if you return home).

This is why the parent insists "displacement, not path length". We needed the vector $\vec{d}$ before that sentence could mean anything.

2. Components — chopping an arrow into $x$ and $y$ pieces

To do arithmetic with arrows we lay down two reference directions: right ($x$) and up ($y$). See Vectors & Components for the full story.

In the figure, drop a straight line from the arrow's head down to the $x$-axis: that ground-shadow is $F_x$ (pink). Drop one across to the $y$-axis: that side-shadow is $F_y$ (blue). The arrow, its $x$-shadow, and its $y$-shadow form a right triangle — a triangle with one $90°$ corner. Hold onto that triangle; the whole $\cos \theta$ story lives inside it.

We write the vector from its components with unit vectors $\hat{ı}$ (one step right) and $\hat{ȷ}$ (one step up): $\vec{F}=F_x\hat{ı}+F_y\hat{ȷ}$ The "hat" means "length exactly 1, just a pure direction". So $3\hat{ı}+4\hat{ȷ}$ means "3 steps right, 4 steps up" — exactly the notation in the parent's Example 2.

3. The angle $\theta$ — and why $\cos$ is the right tool

Now the key question work must answer: how much of $\vec{F}$ points along $\vec{d}$? That "along-$\vec{d}$ shadow" is the piece that rides with the motion. In the figure, drop $\vec{F}$'s shadow onto the direction of $\vec{d}$ (the yellow segment $F_{\parallel }$).

Why cosine and not sine? In the right triangle formed by $\vec{F}$ and its shadow, the shadow $F_{\parallel }$ is the side next to the angle $\theta$ (the adjacent side), and $F$ is the longest side (the hypotenuse). By the very definition of cosine on a right triangle: $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{F_{\parallel }}{F}\Rightarrow F_{\parallel }=F\cos \theta$

So $\cos \theta$ is precisely "what fraction of the full force survives once we point it along the road." Sine would give the perpendicular leftover $F_{\perp }=F\sin \theta$ — the steering part that does no work. Choosing $\cos$ is choosing "along the motion." That is the entire reason the parent's formula is $Fd\cos \theta$ and never $Fd\sin \theta$.

4. The dot product — the two-vectors-in, one-scalar-out machine

Why does the same number have two faces?

• $Fd\cos \theta$ is the geometry face: "(along-force) × (displacement)." Use it when you know the angle.
• $F_x{d}_x+F_y{d}_y$ is the component face: multiply matching shadows, add. Use it when you're handed arrows as $\hat{ı},\hat{ȷ}$ lists and never told $\theta$.

They always agree. Quick check with the parent's Example 2, $\vec{F}=(3,4)$, $\vec{d}=(2,-1)$: $\vec{F}\cdot \vec{d}=(3)(2)+(4)(-1)=6-4=2\text{J}$ The dot product hands you the sign for free — no separate step to decide positive or negative.

5. Units — the joule

Because work is a dot product, its output is a scalar, so a joule carries no direction — only a size and a sign ($+$ energy in, $-$ energy out).

Prerequisite map

Equipment checklist

Cover the right side and test yourself.

Connections

• Parent: Work definition — where these foundations get assembled.
• Vectors & Components — the arrow-chopping this page leans on.
• Dot Product (Scalar Product) — the two-in-one-out machine in full.
• Kinetic Energy & Work-Energy Theorem — where positive/negative work goes next.
• Power — rate of doing work, another dot product $P=\vec{F}\cdot \vec{v}$.
• Friction — a force that reliably does negative work.
• Potential Energy & Conservative Forces — why loop displacement being zero matters.