In $\mathbb{R}^2$ specify a set of vectors, that has many elements as possible, so that no vector is a constant multiple of another vector in the set.
What do such sets have in common?
In $\mathbb{R}^3$ do the same
I asked both of these earlier, but people are acting like it is a HW question (giving me "hints" etc.) but I'm not looking for HW help. I made these questions up.
The answers are the equivalence classes of fractions for $\mathbb{R}^2$ and for $\mathbb{R}^3$ the equivalence classes of 3-part ordered ratios.
But now that you have heard that answer are there any more interesting answers?