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Is the presence of exponentiation relevant here besides an easy way to do multiplication for magnitudes and addition for angles? (i.e. does the use of e as the constant for exponentiation matter for the purposes of these calculations?)


Are there cases where the rotation matrices on the left are better suited to a graphics problem than the polar coordinates on the right? It seems like the right equations are much simpler as you mentioned, but could there still be uses for the left equations?


Looks like e^(i * theta) has two meaning: it either represents a vector with length 1 that rotates theta CCW from x axis, or it means the operation of rotating a vector theta CCW from it current position while maintaining its length.


How does complex number calculations get evaluated in computers?


Another way to prove 2D rotations communicate right?


About precision, is complex computation has more precision than matrix form?


Suppose we do perform a 2D rotation using complex numbers, how would we revert back so that we get a vector with real components?


Is there an advantage of using one coordinate system over the other?


How does our computer use complex coordinates to perform calculations? I'd imagine it would be rather "expensive" to implement so is there a significantly greater benefit for doing so?


Are polar coordinates the reason why the complex system is so nice for 2D?