Consider an ordered-set of three numbers , and that constitute an algebraic vector written as . These numbers can be compressed into a single number called the *norm* of . Evaluating the norm depends on several more numbers that are called components of a metric. The numerical values of these metric components are established by the context of a calculation, they are often implicit.

A complete mathematical discussion of metrics and norms can be quite extended, so for EthnoPhysics we focus on the specific case where and are the three radii used to describe the shape of a particle. These radii are determined by counting quarks, so to assess their norm we use components of the quark metric, written as . First we define a directed **surface** **area** by

In general, radii may be positive or negative, so may be positive or negative too. If we say that the surface of P is **outside** facing. And if , the surface is facing **inside**. The **norm** of is defined as

This number may be imaginary^{1}The term *imaginary* is used here with its mathematical meaning . if P’s surface is facing inward. Note that particles and anti-particles have opposing radius vectors, . And all radii appear as paired factors in the expression for . So both vectors have the same norm, and we write

## The Inner Product

Here is another useful way to distill two radius vectors into a single number. Let us call the vectors and . Then the **inner product** is defined by

We say that and are **orthogonal** if .

1 | The term imaginary is used here with its mathematical meaning . |
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