Multiply and add

Why this matters. Search engines, music recommendations and ChatGPT all lean on one small calculation, the dot product. It is how a computer measures how alike two things are: two songs, two documents, two words. Here it is on arrows you can see, before the end of the lab shows where it lives.

The dot product of two arrows is one number: multiply the x parts, multiply the y parts, add the two results. For u = (3, 1) and v = (2, 5) it is 3·2 + 1·5 = 11.

That number says how much the two arrows agree. Pointing roughly the same way gives a positive number, pointing roughly opposite ways gives a negative one, and at exactly a right angle the two parts cancel and it is 0.

Build three arrows so that u·v comes out as 7, then 0, then −5. Watch where each arrow ends up compared with u.