g.lemaitre said:
Well, I find the book's use of the English language very unfortunate. If I say a group of people is multi-ethic if any two of its members are of a different race, then it is multiethic. That means only two of its members need be different, not all of them. But if orthogonal means that all pairs must perpendicular then that's the way it is.
ehild said:
That is the definition of an orthogonal set of vectors. All possible pairs are orthogonal. Choosing
any two vectors, they are orthogonal. The multi-ethic group means that you can choose at least one pair of people who belong to different races. I would not use "any" in this case.
"multiethic" does not mean multiple races.
The definition you show for orthogonality is incorrect (translation error?). Here's a simple counterexample.
Let S = {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}
Clearly, the first two vectors in the list above are orthogonal, so by the posted definition, the entire set is orthogonal. However, taking dot products, we see that <1, 0, 0>##\cdot## <1, 1, 0> = 1, so these two vectors aren't orthogonal.
Likewise, <0, 1, 0>##\cdot## <1, 1, 0> = 1, so these two vectors aren't orthogonal, either.
For a set of vectors to be orthogonal,
every pair of them must be orthogonal.
