The Norm Nine Dragon Tampan. Paminggir people. Sumatra circa 1900, 41 x 43 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
Consider an ordered-set of three numbers $\rho_{m} \,$, $\rho_{e}$ and $\rho_{z}$ that constitute an that is generically written as

$\overline{\rho} = \left( \, \rho_{m}, \rho_{e}, \rho_{z} \right)$

These three numbers can be compressed into a single number called the norm of $\overline{\rho}$ and usually noted by $\left\| \, \overline{\rho} \, \right\|$. This norm depends on nine constants that are collectively called a metric. They are written using $k$ with a subscript. The numerical values for these constants are established by the context of a calculation, they are often implicit. A complete mathematical discussion of the norm can be quite extended, so for WikiMechanics we focus on the specific case where the numbers $\rho_{e} \,$, $\rho_{m}$ and $\rho_{z}$ are the three radii used to describe the shape of a particle. These radii are determined by counting quarks, so we use the quark metric to assess the norm. Follow the link for more details, but note that the cross-terms are symmetric so $k_{em} = k_{me}$, $k_{ez} = k_{ze}$ and $k_{mz} = k_{zm}$. Then we write the square of the norm as

$\left\| \, \overline{\rho} \, \right\| ^{2} = k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} + 2 k_{em} \rho_{m} \rho_{e} + 2k_{mz}\rho_{m} \rho_{z} + 2 k_{ez}\rho_{e} \rho_{z}$

We consider that this quantity may be positive or negative, but not a complex or imaginary number. So its square root is either all real, or all imaginary. Then we can easily find the norm by taking an absolute value before the square root

$\left\| \, \overline{\rho} \, \right\| \equiv \sqrt{ \vphantom{\sum^{2} } \; \left| \; k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} + 2 k_{em} \rho_{m} \rho_{e} + 2k_{mz}\rho_{m} \rho_{z} + 2 k_{ez}\rho_{e} \rho_{z} \; \right| \; }$

The foregoing can be generalized to define another useful way to distill two radius vectors into a single number. Let us call the vectors $\overline{a} = \left( \, a_{m}, a_{e}, a_{z} \right)$ and $\overline{b} = \left( \, b_{m}, b_{e}, b_{z} \right)$. Then the inner product of these two vectors is defined by

$\overline{a} \bullet \overline{b} = k_{mm} a_{m} b_{m} + k_{ee} a_{e} b_{e} + k_{zz} a_{z} b_{z} + 2 k_{em} a_{m} b_{e} + 2k_{mz} a_{m} b_{z} + 2 k_{ez} a_{e} b_{z}$

We say that $\overline{a}$ and $\overline{b}$ are orthogonal if $\overline{a} \bullet \overline{b} = 0$.

 Summary
 Noun Definition Norm $\hspace{-150px} {\large{\parallel}} \, \overline{\rho} \, {\large{\parallel}} \equiv \sqrt{ \ {\large{\rvert}} \, k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} + 2 k_{me} \rho_{m} \rho_{e} + 2k_{mz}\rho_{m} \rho_{z} + 2 k_{ez}\rho_{e} \rho_{z} \, {\large{\rvert}} \ } \, \, \,$ 5-22
 Noun Definition Inner Product $\hspace{-150px} \overline{a} \bullet \overline{b} = k_{mm} a_{m} b_{m} + k_{ee} a_{e} b_{e} + k_{zz} a_{z} b_{z} + 2 k_{em} a_{m} b_{e} + 2k_{mz} a_{m} b_{z} + 2 k_{ez} a_{e} b_{z}$ 5-23
 Adjective Definition Orthogonal $\overline{a} \bullet \overline{b} = 0$ 5-24
page revision: 415, last edited: 05 Nov 2019 14:30