Somewhat unclear, but that sounds good.
Maybe?
Are you trying to say that the conservation of linear momentum implies the conservation of angular momentum?
In that case, I could buy into that. It might be preferable to show this more rigorously by making arguments that rigid rotators are...
The easiest way to get this is to flip the x and y axes around.
So, you could draw an imaginary line (like a dashed line) along the x-y line (45 degrees from the x-axis in quadrant 1 going through the origin).
Then, take the lines under the dashed line, draw them on top like they'd be seen in...
Right, true. Yes, ok. Fine, but functions don't by definition need to be one-to-one. That's just an injective ("one-to-one") function. Restricting the domain to pass the vertical line test doesn't need to happen to define it as a function...just a one-to-one function.
I thought if...
That's neat! The square root of the imaginary term can be rewritten as
i=e^{i\tfrac{\pi}{2}}
\sqrt{i}=e^{i\tfrac{\pi}{4}}
\sqrt{i}=\cos\tfrac{\pi}{4}+i\sin\tfrac{\pi}{4}
\sqrt{i}=\tfrac{\sqrt{2}}{2}\left(1+i\right)
Ok, we can use the same idea then. See if you can find a number that gets from one line to the other. I wouldn't use a cross product, but if I did, then I'd need to keep the constants.
I'll also hazard that posting sort of opens you up to questions about process and understanding. The...