S17. Geometrical Properties of 3D space

Before dealing with the spacetime geometry, let us first look at the geometry of the three dimensional space. Infact, let us make it even more simplier by looking at the two dimensional axes.

If we consider two points $P_1$ and $P_2$ with coordinates $(x_1,y_1)$ and $(x_2,y_2)$, then the line joining these two points defines a vector $\Delta \vec{r}$. This can be written in the component form in the given coordinate system $S$ as follows,

 Now, the $S$ frame system is not only the coordinate system that we can have, for instance we can have a tilted axes or in other words axes rotated relative to the $S$ about $Z-$axis.

and so in this case, the $S'$ coordinate system frame we have the $\Delta \vec{r'}$

Although the new $\Delta \vec{r'}$ has different primed coordinate, the vector is itself the same. The relation between the two different $\Delta \vec{r}$ and $\Delta \vec{r'}$ coordinate representation is as follows,

Generalizing the rotation about z-axis in 3d space, we have the following,

The main point of this discussion of the transformation of the components from one coordinate to the other coordinate is to realize that although the components of the vectors can have different values, one particular feature remains unchanged "the length". We say that the length is invariant properties of the vector in 3d space.

Using the above transformation rule, we can show that square of the length is infact invariant quantity. Or in general length is invariant under any complicated transformation.  i.e.
                           $\Delta x^2+ \Delta y^2+ \Delta z^2=\Delta x'^2 +\Delta y'^2 +\Delta z'^2$

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