Universal Geometry

Three important descriptions of perpendicularity define three different geometries, named by Dr. Wildberger as red, green, and blue. Blue represents Euclidean geometry, while the red and green are two different but related relativistic (hyperbolic) geometries. Three different definitions of the dot product correspond to these three different geometryies:

$$ \begin{align} (x_1,y_1) \cdot_b (x_2,y_2) &= x_1x_2 + y_1y_2 \\ (x_1,y_1) \cdot_r (x_2,y_2) &= x_1x_2 - y_1y_2 \\ (x_1,y_1) \cdot_g (x_2,y_2) &= x_1y_2 + y_1x_2 \\ \end{align} $$

From a linear algebra perspective, this system can be written using $$1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$, $$i = \begin{bmatrix} 0 & 1 \\ -1 & 0 \\ \end{bmatrix} $$, $$j = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} $$, and $$k = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} $$. (This system is very similar to Hamiltonian quaternions, in that $$i^2=-1$$, but note that $$j^2=k^2=1$$ and yet $$j\ne k\ne 1$$.)  As an aside, it is convenient also to have an infinitesimal, $$\epsilon = \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$$.

In each system, perpendicularity is defined as precisely those situations where $$v\cdot w = 0$$