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Learn physical dimensions as codes built from base quantities, how to derive dimensions of derived quantities, and how to check formulas without numbers.

---

## Dimension Is Not Just Object Size

In everyday language, dimension often means spatial size such as length, width, and height. In physics, **dimension** has a more specific meaning: it is a code that shows which base quantities build a physical quantity.

For example, bolt length, nut diameter, marble radius, and travel distance can use different units, but they are still the same kind of quantity.

```math
\text{length, diameter, radius, distance} \Rightarrow [\mathrm{L}]
```

A dimension does not tell us the measured value. It tells us the *physical kind* behind the measured value.

## Seeing Dimension Powers

The visual below focuses only on the length dimension. Switch from length to area and then volume. Notice how one length factor becomes $$[\mathrm{L}]$$, $$[\mathrm{L}]^2$$, and $$[\mathrm{L}]^3$$.

Visible text: The visual below focuses only on the length dimension. Switch from length to area and then volume. Notice how one length factor becomes , , and .

Component: DimensionLab
Props:
- title: Length Dimension Builder
- description: Choose a shape built from length factors to see its formula, unit, and
dimension.
- labels: {
chooseMode: "Choose a dimension shape",
dimension: "Dimension",
formula: "Formula",
modes: {
length: "Length",
area: "Area",
volume: "Volume",
},
unit: "SI unit",
}

## The Alphabet of Base Quantities

The International System of Units or SI uses $$7$$ base quantities. In dimensional analysis, these base quantities work like an alphabet for building other quantities.

Visible text: The International System of Units or SI uses base quantities. In dimensional analysis, these base quantities work like an alphabet for building other quantities.

| Base quantity | Example symbol | Dimension |
| :------------ | :------------- | :-------- |
| Length | $$\ell,\ x,\ r$$ | $$[\mathrm{L}]$$ |
| Mass | $$m$$ | $$[\mathrm{M}]$$ |
| Time | $$t$$ | $$[\mathrm{T}]$$ |
| Electric current | $$I$$ | $$[\mathrm{I}]$$ |
| Thermodynamic temperature | $$T$$ | $$[\Theta]$$ |
| Amount of substance | $$n$$ | $$[\mathrm{N}]$$ |
| Luminous intensity | $$I_v$$ | $$[\mathrm{J}]$$ |

Visible text: | Base quantity | Example symbol | Dimension |
| :------------ | :------------- | :-------- |
| Length | | |
| Mass | | |
| Time | | |
| Electric current | | |
| Thermodynamic temperature | | |
| Amount of substance | | |
| Luminous intensity | | |

Dimensions are written in square brackets so they are not confused with units. For example, $$[\mathrm{L}]$$ is the dimension of length, while $$\text{m}$$ is the unit meter.

Visible text: Dimensions are written in square brackets so they are not confused with units. For example, is the dimension of length, while is the unit meter.

## Deriving Dimensions from Formulas

Dimensional analysis works by replacing every quantity in a formula with its dimension, then simplifying the powers.

```math
\begin{aligned}
[A] &= [l][w] = [\mathrm{L}][\mathrm{L}] = [\mathrm{L}]^2 \\
[V] &= [l][w][h] = [\mathrm{L}]^3 \\
[v] &= \frac{[\Delta s]}{[\Delta t]} = \frac{[\mathrm{L}]}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-1} \\
[a] &= \frac{[\Delta v]}{[\Delta t]} = \frac{[\mathrm{L}][\mathrm{T}]^{-1}}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-2}
\end{aligned}
```

Force, work, and power can also be read as dimension structures.

```math
\begin{aligned}
[F] &= [m][a] = [\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2} \\
[W] &= [F][\Delta s] = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2} \\
[P] &= \frac{[W]}{[t]} = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}
\end{aligned}
```

## A Formula That Passes Inspection

Dimensions can help check whether a formula form could be correct. Every term being added must have the same dimension.

Take the displacement formula for uniformly accelerated motion:

```math
s = v_0 t + \frac{1}{2} a t^2
```

Check the dimensions:

```math
\begin{aligned}
[v_0t] &= [\mathrm{L}][\mathrm{T}]^{-1}[\mathrm{T}] = [\mathrm{L}] \\
[at^2] &= [\mathrm{L}][\mathrm{T}]^{-2}[\mathrm{T}]^2 = [\mathrm{L}]
\end{aligned}
```

Both terms on the right have dimension $$[\mathrm{L}]$$, so the formula passes the dimensional check. This check does not prove the formula is definitely correct, but it can quickly catch impossible formulas.

Visible text: Both terms on the right have dimension , so the formula passes the dimensional check. This check does not prove the formula is definitely correct, but it can quickly catch impossible formulas.

```math
s + t \quad \text{is not valid because} \quad [\mathrm{L}] \ne [\mathrm{T}]
```

## One Object, Many Dimensions

A bolt and a nut look like small objects, but measuring them can involve several different quantities. That is why one object sometimes needs more than one measuring tool.

| What is checked | Suitable tool | Dimension |
| :-------------- | :------------ | :-------- |
| Bolt length | ruler or vernier caliper | $$[\mathrm{L}]$$ |
| Outer diameter | vernier caliper or micrometer screw gauge | $$[\mathrm{L}]$$ |
| Cross-sectional area | calculated from diameter | $$[\mathrm{L}]^2$$ |
| Material volume | calculated from spatial size | $$[\mathrm{L}]^3$$ |
| Mass | balance | $$[\mathrm{M}]$$ |
| Density | mass divided by volume | $$[\mathrm{M}][\mathrm{L}]^{-3}$$ |

Visible text: | What is checked | Suitable tool | Dimension |
| :-------------- | :------------ | :-------- |
| Bolt length | ruler or vernier caliper | |
| Outer diameter | vernier caliper or micrometer screw gauge | |
| Cross-sectional area | calculated from diameter | |
| Material volume | calculated from spatial size | |
| Mass | balance | |
| Density | mass divided by volume | |

So, two measuring tools can measure quantities with the same dimension, but they serve different contexts. A ruler is enough for a large length that does not need high precision. A vernier caliper or micrometer screw gauge is better when a small diameter must be read more carefully.

OpenStax's concept reference for dimensional analysis can be opened through [this source link](https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis).