Everyone tells me that I should use right-handed coordinate systems. But no one tells me what happens if I don't. What is the danger of not using right-handed coordinate systems?
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What do you mean? How exactly are you going to "use" left-handed coordinate systems?
It's a convention, nothing more. Your signs will be backwards in some calculations.
(The real reason has to do with the fact that the determinant operator is uniquely determined by a basis AND an orientation. If you choose a different orientation than everybody else, your determinant will be the negative of theirs. It's a convention, but it's a convention that really matters. I'm guessing that this doesn't mean anything to you, and you probably just want to label your axes differently when you're graphing stuff. If so, please disregard.)
cross product changes too (related to the change in determinant operator)
many things like electromagnetism etc... motivate the use of right hand system as just a convenient way to present certain physical phenomena. But at the end of the day, it is just matter of convention. nothing else