Undergrad Is the curl of a vector perpendicular to the vector?

[Pushoam]

I am asking this because ∇ is also considered as a vector in some cases. Considering it as a vector in this case ,too,
( ∇× E).E = (- ∂ B/∂t).E = - ∂ (B.E)/∂t =0

Since B and E could be arbitrarily dependent on t,
B.E = 0

where B and E are magnetic and electric fields respectively.
This means that B and E are always perpendicular to each other.

 

[jedishrfu]

I wouldn't consider $\nabla$ a vector, its really an operator with many interesting properties in Vector Calculus. Here's a brief discussion on it:

https://betterexplained.com/articles/vector-calculus-understanding-circulation-and-curl/

Curl was for me somewhat mysterious until I read this article on it and they use the little pinwheel in the flow to determine if there is rotation.