Christian's answer already covers most of what you asked for expect your concrete example from the end of your question about representing "North-East":

First of all, it depends on your coordinate system. But if we look onto your world from the top and assume that north/south is the y-axis (+y = north) and west/east is the x-axis (+x = east), you could represent the direction "North-East" as the (un-normalized) vector [1, 1] or the (normalized] vector:

Example visualization in WolframAlpha.

Christian's answer already covers most of what you asked for expect your concrete example from the end of your question about representing "North-East":

First of all, it depends on your coordinate system. But if we look onto your world from the top and assume that North/South is the y-axis (+y = North) and West/East is the x-axis (+x = East), you could represent the direction "North-East" as the (un-normalized) vector [1, 1] or the (normalized] vector:

Example visualization in WolframAlpha.

Christian's answer already covers most of what you asked for expect your concrete example from the end of your question about representing "North-East":

First of all, it depends on your coordinate system. But if we look onto your world from the top and assume that north/south is the y-axis (+y = north) and west/east is the x-axis (+x = east), you could represent the direction "North-East" as the (un-normalized) vector [1, 1] or the (normalized] vector [1/sqrt(2), 1/sqrt(2)].

Example visualization in WolframAlpha.

Christian's answer already covers most of what you asked for expect your concrete example from the end of your question about representing "North-East":

First of all, it depends on your coordinate system. But if we look onto your world from the top and assume that north/south is the y-axis (+y = north) and west/east is the x-axis (+x = east), you could represent the direction "North-East" as the (un-normalized) vector [1, 1] or the (normalized] vector:

Example visualization in WolframAlpha.