Vectors are arrows

Why this matters. Every photo you rotate or resize on a phone, every 3D game turning its camera, every map tilting as you walk is a matrix moving points, exactly as you are about to. Machine learning runs on the same idea: its data is a big table of numbers, and matrices are how that data gets moved and combined. This lab starts with the smallest case, two numbers and an arrow.

A vector is an arrow from the origin: two numbers, an x step and a y step. Drawn on the grid below, u and v are both arrows starting at (0, 0).

Adding two vectors means placing the second one's tail at the first one's tip — u then v, one after another — and the sum is wherever that lands you. This is why it's called "tip to tail": you never add the numbers in your head first, you just walk the two arrows.