When the Curl of a Vector Field is Orthogonal

[Harrisonized]

Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't seen contrary in Griffifth's so far.) The reason he gave is that if you dot A into the curl of A, you'll end up taking the determinant of matrix with two of the same rows. Therefore, that determinant is 0. Since that is the equivalent of taking the dot product of A and the curl of A, and the curl of A is B, then A and B are orthogonal because their dot product is zero, and only orthogonal vectors give a dot product of 0.

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?

 

[TSny]

Doesn't sound right to me. You could add arbitrary constants to the components of A without changing the curl, but these constants will change the direction of A.

For a specific example, Let A = y##\hat{i}## + ##\hat{k}##