Euclidean Foundations: Simple Criterion for Orthogonality of Direction Vectors

Great, so we want the axes $\{(tx,ty)\}$ and $\{(-ty,tx)\}$ to be orthogonal.

To formalize this, it'll be easier to define orthogonal vectors/directions instead. Intuitively, and ignoring bad division by zero cases, vectors $(a,b)$ and $(c,d)$ should be orthogonal if for some nonzero vector $(x,y)$, we have $b/a = y/x$ and $d/c = x/(-y)$, which is basically equivalent to $b/a = -c/d$, or more simply, $ac+bd = 0$.

This will be our definition: two vectors $(a,b)$ and $(c,d)$ are orthogonal iff $ac+bd = 0$.

Similarly, two lines/rays are orthogonal iff their direction vectors are orthogonal.