Let
be a family of independent
-dimensional standard Gaussian random variables (or any other continuous distribution supported by
). Almost surely, this family answers your question.
Density: comes as a direct consequence of Borel-Cantelli lemma (for any ball
,
hence
, and these events are independent, so that infinitely many terms of the sequence fall in
almost-surely)
Non-colinearity: for any distinct
,
because the line
has zero Lebesgue measure (since
), and
has a density, and is independent of
. There are countably many triplets
, hence the probability that there exists one satisfying the alignment property is 0. (A union of countably many sets of zero measure has zero measure).