Module 02

Coordinates and vectors

6 minute read

A single arrow is painted on a rug. You stand at one corner; a friend stands at another, turned some angle from you. Describe the arrow, you are both told, and be exact.

You say: from its tail, go three steps toward the window and four steps toward the bookshelf. Your friend says: nonsense. From its tail, go five steps straight ahead. You compare notes and find you disagree about every single number. You walk the rug together, measure twice, and discover something worse: you are both right.

The arrow did not move. Nobody lied. Yet the numbers describing it refuse to match. So: if two honest observers cannot agree on the numbers, what exactly is real about the arrow?

The takeaway

The arrow is real; the numbers are a description written in somebody's grid. Physics lives in the parts of the description that every grid agrees on.

The grid is yours, the arrow is not

Here is what you actually did on that rug. You laid an invisible sheet of graph paper over the world, with its lines running window-ward and bookshelf-ward, and you read the arrow off your paper. Your friend laid down different paper, turned relative to yours, and read the same arrow off theirs. The paper is called a frame, the ruled directions are coordinates, and the two numbers each of you read off are components. The graph-paper picture is an analogy, and here is where it breaks: real paper exists, while a frame costs nothing and changes nothing. Laying a new one over the world is a decision, not an event. Nothing physical happens to the arrow when you re-grid it.

The arrow itself, the thing both descriptions point at, is a vector: a length together with a direction. Displacement is the archetype. Go that way, this far. The most useful fact in this module is a split running through everything you will ever compute: some numbers belong to the arrow, and some belong to the paper. Components belong to the paper. Turn the paper and they change. The arrow's length, its magnitude, belongs to the arrow. Turn the paper all you like; five steps is five steps.

Be careful with the word, because not every pair of numbers earns it. The temperature and the air pressure in your kitchen make a perfectly good pair of numbers, and they are not a vector, because nothing about them points anywhere. Turn your graph paper and the temperature could not care less. The test for vectorhood is exactly the rug experiment: measure the thing in one frame, turn the frame, measure again. If the numbers reshuffle according to the geometry of the turn, arrow-style, you have a vector. If the numbers sit still, you have a plain quantity wearing a costume. Physics uses both kinds constantly and never confuses them, and once you see the split you will not confuse them either.

One more habit, while the rug is out. A vector does not live at a place. The arrow from the rug's corner and the same-length, same-direction arrow across town are the same vector; displacement cares about the trip, not the address. This sounds like pedantry until module 06, where whole collections of arrows get added that were measured at different spots. It works because the arrows are portable.

Figure 2.1 The arrow is identical in both panels; only the paper underneath it turned. The components changed. The length did not.

Why does a course on prediction spend its second sitting on stationery? Because the demon from module 01 has a bookkeeping problem. It must write the state of the world down before it can compute with it, and any writing happens in some frame, and the demon's ledger must not depend on which one it happened to grab. The quantities worth building laws from are the ones that survive a change of paper. That single idea, pushed hard enough, will hand you conservation laws in module 09. For now, hold the modest version: before trusting any number, ask whose grid it came from.

The math

Work on a flat plane. Fix a frame. Write a vector as its two components in that frame:

v = (v_{x}, v_{y})

Add vectors component by component. Scale a vector by scaling each component:

u + v = (u_{x} + v_{x},; u_{y} + v_{y}), c v = (c v_{x},; c v_{y})

The magnitude comes from the Pythagorean theorem:

∣ v ∣ = v_{x}^{2} + v_{y}^{2}

Now rotate the frame by an angle . The same vector gets new components:

v_{x}^{'} = v_{x} cos θ + v_{y} sin θ, v_{y}^{'} = - v_{x} sin θ + v_{y} cos θ

Square both new components. Add them. Expand the squares:

v_{x}^{'2} + v_{y}^{'2} = (v_{x}^{2} + v_{y}^{2}) (cos^{2} θ + sin^{2} θ) = v_{x}^{2} + v_{y}^{2}

The cross terms cancel. The identity removes entirely.

Read the result. The angle is gone. No choice of frame appears in . The magnitude is frame-proof. Quantities with this property are called invariant, and they are the currency this course trades in.

Play with it

One arrow, two frames. Drag the tip anywhere. Turn your friend's frame with the slider. Watch which numbers panic and which one refuses to move.

Check yourself

Problem 1. An arrow has components (3, 4). What is its length? Now rotate the grid a quarter turn, so the old up-direction becomes the new right-direction. What are the new components, and what is the new length?

Reveal the solution

Length: the square root of 9 + 16, which is 5.

A quarter turn sends the old up-axis to the new right-axis, so the component 4 moves to the first slot and the component 3 picks up a sign: the new components are (4, -3). Check it against the rotation formula with theta equal to 90 degrees. The new length is the square root of 16 + 9. Still 5.

Problem 2. You make two displacements in a row: first (1, 2), then (3, -2). What single displacement takes you from start to finish, and how long is it?

Reveal the solution

Add component by component: (1 + 3, 2 - 2) = (4, 0). Its length is 4. The 2 up and the 2 down cancel; the whole trip lands you four units along the first axis.

Problem 3. You measure an arrow as (3, 4). Your friend measures the same arrow as (5, 0). Is that possible? If so, roughly what angle separates your two grids?

Reveal the solution

Possible. The test is the invariant: your length is 5 and your friend's length is 5, so both readings can describe one arrow.

Your friend's grid has its first axis pointing along the arrow itself. The arrow sits at the angle whose tangent is 4 over 3 in your grid, about 53 degrees. That is the angle between the grids.

What this did to the demon

The demon now writes its ledger in frame-proof ink: record what survives the grid, translate the rest on demand. But a ledger of snapshots is still an album, not a prediction, because prediction needs the rate at which one snapshot becomes the next. Module 03 builds the instrument that reads change at a single instant.

Next · module 03
The calculus you need