# Usage of the word "orthogonal" outside of mathematics

From the roots ortho (straight) and gon (angle), its meaning in mathematics is understandable. Outside of mathematics it has various meanings depending on the context:

• Debate - orthogonal: not relevant
• Statistics - orthogonal: unrelated
• Computing - orthogonal: isolated or partitioned

There are other definitions. Most seem to imply a meaning of independence or separation. Does anyone know how it came to mean this?

If you think about (simplified for convenience) mathematical usage of "orthogonal", it is referring to vectors at right angles to each other, so motion in the direction of the first vector produces no corresponding motion in the direction of the second vector. This independence is what motivates the other meanings; an orthogonal line of argument in debate might be interesting in itself, but doesn't advance the main thrust of the debate, for instance.

It's just as well that the mathematical use of "normal" doesn't bleed across like this, because "normal people" would then be at right-angles to reality. Then again... :-)

• To extend this beyond debates, two political issues are orthogonal if moving on one won't change your position on the other. Commented Jul 28, 2017 at 20:32
• I thought "normal" was related to the number 1 in math. Examples would include an orthonormal basis consisting of vectors of length 1, and the gradients of 2 normal lines multiplying to equal -1. Since we usually use things of length 1, "normal" can be this way associated with "usual things". Commented May 25, 2018 at 1:17
• `"normal people" would then be at right-angles to reality. ` That made my day. Commented Jul 18, 2022 at 12:02
• I always found the use of orthogonal outside of mathematics to confuse conversation. You might imagine two orthogonal lines or topics intersecting perfecting and deriving meaning from that symbolize Commented Oct 25, 2022 at 3:49

As Wikipedia says about the derived meanings of orthogonal, they all "evolved from its earlier use in mathematics".

• In statistics, the meaning of orthogonal as unrelated (or more precisely uncorrelated) is very directly related to the mathematical definition. [Two vectors x and y are called orthogonal if the projection of x in the direction of y (or vice-versa) is zero; this is geometrically the same as being at right angles.]
The statistical meaning comes exactly from this: one can think of random variables as living in a vector space, and correlation between two random variables is zero precisely when the two vectors are orthogonal/"perpendicular". See this post for details.

• In debate(?), "orthogonal" to mean "not relevant" or "unrelated" also comes from the above meaning. If issues X and Y are "orthogonal", then X has no bearing on Y. If you think of X and Y as vectors, then X has no component in the direction of Y: in other words, it is orthogonal in the mathematical sense.

• In computing, the use of orthogonal for isolated or partitioned (which I don't actually recall encountering) would come from the same meaning: the behaviour of one component has no bearing on (is isolated from) other components; so they are orthogonal.

• "linearly independent" doesn't mean "orthogonal". Orthogonality implies linear independence, but the converse is not true. Commented Feb 11, 2011 at 21:43
• @Peter: Thanks; that was very stupid of me. :-) I've fixed it now. Commented Feb 12, 2011 at 6:27

because in mathematics orthogonal is synonym of being independent or absolute lack of dependence.There are intermediate states from no dependence to complete dependence (aka parallel) which is given by vector product

Update: I did not want to implicate but without it my answer is not answering direct question.

In Russian primary school the vectors are studied in the 4-7th form and after that children tell in conversations "parallel" instead of "dependent and "orthogonal" (or "perpendicular") instead of "independent".

• As Peter pointed out, orthogonal implies but is not a synonym of being independent. For instance, in the plane (two dimensions), any two vectors that are not parallel are actually independent (i.e., they are not dependent; no linear combination of them is zero; one cannot be expressed in terms of the other), but orthogonal only when they are perpendicular. Commented Feb 12, 2011 at 6:31
• Independant vector X in relation to vector set {Y} supposes that it cannot be equal to a linear combination of vector set {Y}. If set contains only one vector, that means it cannot be a collinear vector of different metric (length). Orthogonal vectors are independant, but they cannot be represented as linear combination of mutually dependant vectors Commented Jul 18, 2022 at 12:09

In construction fields, orthogonal is used instead of perpendicular.

There seems to be another sense of orthogonal as "orthogonal categories" eg suppose we have two sets of categories I {A, B,..} and II {C, D,...} then to claim " I and II are orthogonal" seems to require the existence of: A-C, A-D,... and B-C , B-D ...; this interests me because I have a problem , where I am uncertain whether I and II are "orthogonal" or "non-orthogonal". Non-orthogonal would seem to imply A==C and B==D etc but is this unique or would non-orthogonal also include the possibilities A==D , B==D etc

• This seems like a different question..
– JJJ
Commented May 26, 2018 at 6:39
• This does seem to be the same sense as in an Orthogonal instruction set. Commented May 26, 2018 at 21:18