/sci/ - Science & Math

Orthogonality of vectors in 2-space is defined as
(x,y) = 0
whereas in 3-space, to show mutual orthogonality, you must show
(x,y) = 0 & (x,z) = 0 & (y,z) = 0

I wanted to know... Is there a way to prove 3-space mutual orthogonality without splitting that into a proof of orthogonality in 3-1=2 space?

Or is there a separate definition of "orthogonality" altogether for N-space higher than 2?