We usually represent vectors in R2 with respect to the standard basis so vectors in the first quadrant have both coordinates positive.

Moving counterclockwise around the origin, we cycle thru four regions:

Using this basis

gives the same counterclockwise cycle. We say these two bases have the same orientation.
(a) Why do they give the same cycle?
(b) What other configurations of unit vectors on the axes give the same cycle?
(c) Find the determinants of the matrices formed from those (ordered) bases.
(d) What other counterclockwise cycles are possible, and what are the associated determinants?
(e) What happens in R1?
(f) What happens in R3?

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