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When I learned about the curl back in vector calculus, I was never told any of this. I can't even find in my book where it says that the curl is orthogonal to itself. This is because it's not. A simple google search gave me counterexamples of when the curl is not orthogonal to itself.

However, my question is this: when is the curl orthogonal to the original vector field? Will I ever see such situations in my E/M class?