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My intuition tells me that: Given a divergence free vector field [itex] \mathbf{F} [/itex], then the curl of the field will be perpendicular to field.

But I'm having a hard time proving this to my self.

I'know that : [itex] \nabla\cdot\mathbf{F} = 0 \hspace{3mm} \Rightarrow \hspace{3mm} \exists\mathbf{A}: \mathbf{F} = \nabla\times\mathbf{A} [/itex]

Therefore : [itex] \mathbf{F}\cdot(\nabla\times\mathbf{F}) = 0 \hspace{3mm} \Rightarrow \hspace{3mm} [\nabla\times\mathbf{A}]\cdot[\nabla\times(\nabla\times\mathbf{A})] = 0 [/itex]

But I can't prove that this actually equals zero... Please help!!