Mathematics / Lesson 03
Vectors and coordinate change
Separate a geometric direction from the coordinates used to describe it.
orient
Why this idea had to exist
Directions survive changes of viewpoint. A displacement across a room is the same physical move whether the room is measured from the north wall or the doorway. Vectors let the move remain stable while coordinates change.
intuition
Build a picture you can reason with
Treat a vector as an instruction arrow. Coordinates are a receipt describing how much of each chosen basis arrow was used. Rotate the basis and the receipt changes, but the instruction arrow does not.
formalize
Give the intuition a precise edge
A vector space supports addition and scalar multiplication. Relative to basis vectors e₁ and e₂, v=v₁e₁+v₂e₂. Its Euclidean magnitude is √(v₁²+v₂²). A change-of-basis matrix rewrites coordinates while preserving the represented vector.
work through
Follow the decisions, not just the symbols
Add v=(3,1) and w=(-1,2) component by component to obtain (2,3). This is not a memorized trick: joining the arrows head-to-tail produces the same final displacement as combining each basis contribution.
experiment
Change one thing and watch the model answer
Compose two draggable arrows and compare component addition with the head-to-tail construction. Then rotate the coordinate axes and watch the coordinates move while the resultant arrow stays fixed.
retrieve
Close the page and reconstruct it
Answer before opening the explanation. Retrieval is evidence only when the answer is produced without a hint.
What remains unchanged when a vector is written in a new basis?
The geometric vector remains unchanged; only its coordinate representation changes.
transfer
Move the idea into a new setting
Model a boat crossing a river by adding its velocity relative to the water to the water's velocity relative to the bank.
reflect
Leave with a diagnostic habit
Whenever coordinates look mysterious, ask what basis they refer to and what underlying object should remain invariant.
Source record
Follow the idea back.
- Reference
- Algebra and Trigonometry 2e
- Publisher
- OpenStax
- License
- CC BY 4.0
- Accessed
- 2026-08-12
- URL
- https://openstax.org/details/books/algebra-and-trigonometry-2e