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Coordinate systems and basis changes

Coordinates are a description, not the object being described. Rotate the reference axes and the same physical vector gets new component values.

Think like a programmer

Components are serialized relative to a basis. Changing basis is a transformation of representation, like reading the same byte sequence with a different agreed format—not a change to the physical vector itself.

Model checklist

Inputs
A physical vector and a basis rotation angle.
State
The vector components in the original basis.
Rule
Apply an inverse rotation to express components in the rotated basis.
Output
New component values.
Check
Vector magnitude is unchanged by rotation.
\[\\begin{bmatrix}v'_x\\\\v'_y\\end{bmatrix}=\\begin{bmatrix}\\cos\\theta&\\sin\\theta\\\\-\\sin\\theta&\\cos\\theta\\end{bmatrix}\\begin{bmatrix}v_x\\\\v_y\\end{bmatrix}\]

A fixed arrow, a rotating basis

Rotate the coordinate axes. The physical arrow stays fixed; its stored components change in the new basis.

Basis rotated 0°. Components in the rotated basis: [4.00, 2.00].

Try this experiment

Prediction: At a 90-degree basis rotation, the components exchange roles with a sign change.

Set the basis rotation to 90°. Compare the original [4, 2] components with the displayed result and verify that the vector's magnitude has not changed.

Where this model breaks

Rotation alone does not cover moving frames, curved coordinates, or relativistic reference frames. Always name the basis and origin attached to recorded components.

Summary

Separate an object from its coordinate representation. Transform components when the basis changes, then test a geometric invariant such as magnitude.

Glossary

Self-check

  1. Does rotating a basis rotate the physical vector?
  2. Which invariant remains unchanged?
  3. Why should stored vector data name its basis?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Coordinate Systems and Basis Changes, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Vectors and Coordinate Systems, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Coordinate Systems and Basis Changes as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Coordinate Systems and Basis Changes into a test

Rotate a coordinate basis while keeping a physical vector fixed, then see why its components must change.

  1. Name the inputs and units that the vectors and coordinate systems model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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