Norms are for dealing with magnitude not direction (and were brought up by my parent commenter). If you care about direction specify an angle, quadrant, cone or

Norms are for dealing with magnitude not direction (and were brought up by my parent commenter). If you care about direction specify an angle, quadrant, cone or other subregion that allows you to take the limit to infinity which then doesn’t depend on the norm. Note this is the same in the univariate case where we talk about left and right tail if we need to distinguish.