Yes, in a Hilbert space (i.e. an abstract vector space with an inner product), the definition of orthogonality is that the inner product of two nonzero vectors is zero.
I'm not sure I really know what you mean by geometric vectors.
I suspected trolling in your question about geometric vectors.
Sides of a triangle and elements of your algebra are different domains. In order to translate results between them one needs to prove this makes sense.
In the article the author only shows that inner product of c by itself equals to sum of inner product squires of a and b, if a and b are orthogonal.
Who told you this has anything to do with lengths of triangle sides?
Yes, in a Hilbert space (i.e. an abstract vector space with an inner product), the definition of orthogonality is that the inner product of two nonzero vectors is zero.
I'm not sure I really know what you mean by geometric vectors.