Fundamentals of Time and Relativity

Vectors

In a given spatial 3d Cartesian reference frame, the standard basis vectors

2.1

e_{x} = (1,0,0), e_{y} = (0,1,0), e_{z} = (0,0,1)

point along the three chosen coordinate axes. A line from the origin to a point $P$ with coordinates $(x, y, z)$ then has the vector representation

2.2

x = x e_{x} + y e_{y} + z e_{z}

In Euclidean space the inner product of any two vectors is defined as

2.3

u \cdot v = u_{x} v_{x} + u_{y} v_{y} + u_{z} v_{z}

Obviously then the base vectors satisfy the orthonormality relation

2.4

e_{i} \cdot e_{j} = δ_{i j}, i, j = x, y, z

and the length of a vector is given by the Pythagorean formula:

2.5

| u | = \sqrt{u \cdot u} = \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}}

A Cartesian coordinate system treats the three directions $x, y, z$ in a symmetric fashion. For this reason, a Cartesian system can be rotated, and the same form of the general distance function is maintained in the rotated system.

One can define alternative, non-Cartesian, coordinate systems for an Euclidean space; for instance, cylindrical and spherical coordinate systems are very useful in physics.

The crucial feature of the concept of distance in physics is that distance between points in ordinary space (or between two events that occur at the same moment of time) is a physical invariant: it does not change with the choice of coordinate system.
The numerical formulas for distance in alternative coordinate systems appear to be quite different from the Cartesian formula. But they are defined to give the same results for the distances between physical points.