Terms of art: Orthogonal

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When engineers speak, we sometimes have vocabulary diffusion from one situation to another. These slips of phrases aren’t arbitrary; they make sense, but only if you know where the term comes from originally.

In this series of posts, I’ll attempt to explain one phrase you may have heard an engineer say before. (Many of these terms may be computer science-related, but I’ve caught myself using all of them around non-technical people.)

For our first installment of “terms of art:”

orthogonal |ɔrˈθɑgənl| adjective

The art

a fancy word for perpendicular; extends to other technical fields beyond math : these two lines are orthogonal.

Out of context

  1. at odds with each other; incompatible : the company’s forceful slogan was orthogonal to its otherwise wholesome branding.
  2. fundamentally different : this bug is orthogonal to the first issue.

Agree or disagree in the comments. Feel free to also suggest future terms of art!

P.S.: Since “orthogonal” means “right-angled,” does that mean an “orthogon” is a fancy word for “rectangle?”

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I tend to use "orthogonal" to mean "independent" or "unrelated" or "non-interfering." Those synonyms are definitely in the same vein as "different," but I think there are some subtle distinctions, and it's always good to have a larger set of synonyms, right?
A better technical definition of orthogonal is simply that the inner product between two entities is zero. So in Euclidean geometry the inner product is the dot product and we obviously associate a zero dot product with perpendicular lines. In mathematics (particularly Fourier Analysis/PDE's) and physics (quantum mechanics) orthogonality places a central role in spaces that might not have such an apparent geometrical interpretation. For example the functions Sin[n*Pi*x/L] (n is an integer > 0) are orthogonal with respect to the inner product Integral[Sin[n*Pi*x/L]*Sin[m*Pi*x/L],{x,0,L}] where n is not equal to m. Of course after we see that the inner product between any two of these functions yields zero we can make up a nice geometrical interpretation that these functions form "right-angled" basis functions for an infinite dimensional Hilbert space.