Everything Is a Vector
A vector is a list of numbers — a point in space. In machine learning, that space is where meaning lives. A word, an image, a user, a document: each becomes a vector, and similarity becomes distance.
The dot product
Multiply matching entries and add them up. That is the dot product, and when the vectors are normalised to length one, it is exactly the cosine of the angle between them. Aligned vectors score near 1; opposite ones near -1; unrelated ones near 0.
Matrix = many dot products at once
Stack a lot of vectors side by side and you get a matrix. Multiplying a vector by a matrix computes its dot product against every row simultaneously. This is why hardware vendors fight over matrix multiplication: it is where nearly all the arithmetic in a neural network happens. A whole network is, at bottom, a long chain of matrices with simple nonlinearities in between.
The geometry of meaning
Once words are points, meaning becomes arithmetic. Analogies fall out as vector
offsets: king − man + woman lands near queen. You do not
program that relationship. It emerges from how the vectors were learned.