Etymology of the word "normal" (perpendicular)

Question

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"?

Answer 1

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.

Answer 2

Latin normalis stems from (latin) norma - a tool with a right angle, a carpenter's square.