I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar.

1. What is the rational behind the definition above? I guess, it has got something to do with minimum overlap between $x$ and $y$.

2. Is this unique generalization of the concept of orthogonality from inner product spaces?

Thank you.