Linear algebra · 01 · Arrows that add · 8 min
Vectors
A vector is one object with two faces: an arrow (with direction and length) and a list of numbers (its coordinates). Linear algebra begins the moment you can flip between the two without thinking.
Build the intuition
Arrow and list, same thing
The arrow “3 right, 2 up” and the list (3, 2) carry identical information. The arrow view gives intuition (direction! length!), the list view gives computation (just add the numbers). Physicists draw; computers list; you'll do both.
Adding: tip to tail
Walk along vector a, then along vector b: your total journey is a + b. In lists, simply add matching slots: (3,2) + (−1,2) = (2,4). Displacements, forces, and velocities genuinely combine this way — the math copies reality.
Scaling: same direction, new length
2v is v doubled in length, same heading; −v is v reversed. Scaling and adding are the only two moves in linear algebra — everything else in the subject is these two moves, composed cleverly. (That austerity is why it scales to a million dimensions.)
See it move
Steer two arrows and watch their sum complete the parallelogram. Tip-to-tail isn't a rule to learn — it's a walk to take.
A worked example
The river crossing
Your kayak paddles 4 km/h north; the river flows 3 km/h east. What's your actual motion?
Add the vectors:
Actual speed is the arrow's length:
You travel at 5 km/h, angled downstream — vector addition predicted your real path.
Out in the world
Words as arrows
Language AIs store each word as a vector with hundreds of slots. Directions become meaning: the famous demo king − man + woman lands near queen. Vector arithmetic on ideas — and it's the foundation of every modern chatbot.
Common confusion, cleared
“A vector is a moving thing.”
A vector is a quantity with direction and size — motion is one example. A force, a color (R,G,B), a survey response can all be vectors while sitting perfectly still.
“Vectors live only in 2D or 3D.”
A list of 500 numbers is a 500-dimensional vector, and the rules don't change. You can't picture it; the algebra doesn't care.
Recap
- Vector = arrow = list of coordinates; switch views freely.
- Add slot-by-slot (tip to tail); scale to stretch, shrink, or reverse.
- Those two moves are the entire foundation of the subject.