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Math · Linear algebra

Vector Lab

Build vectors, turn dot products into projections and transform a cube with a matrix you can edit.

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The course

7 worlds · 25 lessons

World 1 · Vectors in space

FREEDirection, length and the plane between two arrows
  1. 01Tip to tail
  2. 02A right angle without a protractor
  3. 03An arrow out of a plane
  4. 04The shadow of an arrow

World 2 · Matrices move space

PROBasis vectors, signed volume and invariant directions
  1. 01A matrix is three new basis vectorsPRO
  2. 02Tilt without changing volumePRO
  3. 03Inside outPRO
  4. 04Squashing a dimensionPRO
  5. 05Find a direction that stays on its linePRO

World 3 · Vectors that change with time

PRORK4, numerical error and the Lorenz system
  1. 01Small steps along a vector fieldPRO
  2. 02Close starts can separatePRO

World 4 · Lines and planes

PRO
  1. 01A line is a point plus a directionPRO
  2. 02Solve where a line meets a planePRO
  3. 03No crossing, or infinitely manyPRO

World 5 · Stability in the phase plane

PRO
  1. 01Two decaying directionsPRO
  2. 02One calm path cannot prove stabilityPRO
  3. 03Stable does not always mean attractingPRO

World 6 · Surfaces: twists, holes and curvature

PROMöbius strip, Klein bottle and Gaussian curvature
  1. 01One side or two?Carry a normal around a loop · 9 minPRO
  2. 02Bending you can measure from insideGaussian curvature K · 9 minPRO
  3. 03Cut the Möbius stripFollow the scissors · 9 minPRO
  4. 04Total curvature counts holesEuler characteristic and Gauss–Bonnet · 9 minPRO

World 7 · Special relativity: spacetime as a matrix

PROLorentz boosts, light cones, slow clocks and short rods
  1. 01Light cones and the order of eventsThe interval and the relativity of simultaneity · 9 minPRO
  2. 02A moving clock runs slowThe light clock and γ · 9 minPRO
  3. 03A moving rod is shorterLength contraction L = L₀/γ · 9 minPRO
  4. 04The twin paradoxProper time along two worldlines · 9 minPRO
Read first

Guides for Vector Lab

All guides →
Exam prep
Forces & MotionAP Physics 1 Forces and Motion: A Free Study GuideMap the forces and motion topics of an introductory physics course to free interactive lessons, with a four-week study plan and tips for the practice exams.4 min read
Explainer
Forces & MotionHow Free Fall and Projectile Motion Work: KinematicsWhy falling speed grows steadily, why distance grows with time squared, and how splitting a throw into two parts predicts range and height. With a live physics lab.5 min read
Explainer
Simple MachinesHow Levers Work: Fulcrum, Torque and Mechanical AdvantageUnderstand the three classes of lever, why a longer arm gives more lifting force, and the balance rule that ties it together. With a live simple-machines lab.4 min read