inner product

An inner product is the grown-up, general version of the dot product: any rule that pairs two vectors and hands back a single number, as long as it obeys a few sensible requirements. Symmetry (the order does not matter), linearity (it plays nicely with scaling and adding), and positivity (a vector paired with itself is never negative, and is zero only for the zero vector).

Those requirements are exactly what is needed to talk about length and angle in a space. From an inner product you define length as the square root of a vector paired with itself, and you call two vectors perpendicular when their inner product is zero.

The familiar dot product is simply the standard inner product on R^n. But other spaces (like spaces of functions) get their own inner products, and the same geometric intuition carries straight over.