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Consider a tuple of distinct positive integers of length $n : [a1, a2, a3,\cdots, an]$.

If there exists an distinct integer tuple of integers of the same length $n : [x1, x2, x3,\cdots, xn]$. such that their dot product is 0. Then what is the term used to denote such integer tuple? Are there any tuples that does not have another tuple such that their dot product is 0?

hanugm
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1 Answers1

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I would say that the tuple $ x $ is perpendicular to $ a $.

In general, ignoring tuples with all zero elements, $ x = (an, an, an,...-\sum_{i=1}^{n-1} ai) $ is an integer tuple solution for $ a.x = 0 $ for any integer tuple $ a = (a1, a2, a3,...an) $. There are, of course, an infinite number of other integer tuple $ x $s also perpendicular to $ a$.

And just for fun, for any integer tuple $ A0 $, you can always find N-1 integer tuples $ A1,A2,A3 $ which are all perpendicular to $ A0 $ and are also all mutually perpendicular to each other.

Penguino
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