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While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or "normal matrix" (which has a set of mutually perpendicular eigenvectors).

However, the word "normal" does not seem to mean its normal/ordinary/usual meaning in English. So, why is a normal vector or matrix called "normal"?

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  • $\begingroup$ The words 'normal' and 'perpendicular' are rarely used in contexts other than $\mathbb{R}^2$ or $\mathbb{R}^3$. These words are replaced by orthogonal, which is a generalization of these ideas in 3 dimensional Euclidean space. $\endgroup$ – noobProgrammer Mar 12 '13 at 18:32
  • $\begingroup$ @noobProgrammer Normal is definitely used in higher-dimensional contexts. Orthogonality is a global concept, while normalness is relative to the surface of something in the space. In otherwords, orthogonality to the tangent vectors at a point when the origin of the space is the point in question. $\endgroup$ – Loki Clock Mar 12 '13 at 18:40
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    $\begingroup$ @noobProgrammer The funny thing is, we have "orthonormal" vectors apart from "orthogonal" vectors, although the "normal" in "orthonormal" means "conforming to the same stardard" (each having a unit length) rather than orthogonality. $\endgroup$ – user1551 Mar 12 '13 at 19:27
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From the OED Third Edition, an entry updated December 2003:

Etymology: < classical Latin normālis right-angled, in post-classical Latin also conforming to or governed by a rule (4th–5th cent.) < norma + -ālis. Compare French normal (1450–65 in Middle French in an isolated attestation in verbe normal (compare sense A. 1), then from mid 18th cent., earliest in ligne normale (1753; compare sense A. 5), and subsequently in more general senses ‘which serves as a model’ (1793 in école normale, 1803 in more general use), ‘ordinary, regular’ (1830s); compare earlier anormal adj.), Italian normale according to the norm, routine, predictable, common, boring (1683 in sense ‘perpendicular, orthogonal’, 1831 in sense ‘customary, expected’), Portuguese normal (1844), Spanish normal (1855).

In short, it goes back to the Latin source of the word.

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    $\begingroup$ Thanks! I've checked Cambridge and Webster, but found no etymological explanation and so I gave up. Never thought that the modern meaning of the word (usual, ordinary) is so modern but the mathematical meaning goes so far back in history. $\endgroup$ – user1551 Mar 12 '13 at 19:19
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    $\begingroup$ @user1551: You’re welcome! A good place to look for etymological information is the Online Etymological Dictionary; it’s an amateur production, but a good one using good sources. $\endgroup$ – Brian M. Scott Mar 12 '13 at 19:23
  • $\begingroup$ That looks useful. It has a long list of print sources (including the Oxford dict.), so I guess it's credible. Thanks again. $\endgroup$ – user1551 Mar 12 '13 at 19:34
  • $\begingroup$ @user1551: It’s not perfect, but neither is any other source. Linguistics, especially historical linguistics, has been a serious hobby of mine for pushing 30 years now, and I’ve found very few out and out errors. $\endgroup$ – Brian M. Scott Mar 12 '13 at 19:37
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Latin normalis stems from (latin) norma - a tool with a right angle, a carpenter's square.

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  • $\begingroup$ Thanks for the answer. That gives a very concrete use case of the word. $\endgroup$ – user1551 Mar 12 '13 at 19:22
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    $\begingroup$ @user1551: The carpenter's square is the original meaning of the word, and demonstrates both the concrete meaning ("at right angles to") and the abstract meaning ("as it ought to be"). $\endgroup$ – MJD Mar 12 '13 at 19:36
  • $\begingroup$ @MJD You're right. I didn't see it. Thanks. $\endgroup$ – user1551 Mar 12 '13 at 20:03

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