Instrument 8 · norms and the dot product

How far is it, really

"Distance" is not one thing. Change the rule for measuring it and the shape of everything equidistant from a point changes with it — a diamond, a circle, a square. Every recommendation engine, every clustering algorithm and every "these two are similar" is a choice about which of those shapes you meant.

Everything at distance 1

How alike are two things

p = 1 · taxi p = 2 · crow p = yours the projection

The same statement, with the numbers in it

{‖ x ‖}_{p} = {(\sum_{i} {| x_{i} |}^{p})}^{1 / p}

{‖ (3, 4) ‖}_{p} = 5.000, \cos θ = 0.000

At p = 2 this is Pythagoras — the straight line. At p = 1 you must travel along the grid, like a taxi. As p grows the rule approaches "whichever coordinate is biggest", and the circle becomes a square. Same point, different distances.

Fine tune

Which rule of distance 2

The angle between two things 60°