How do I find orthogonal vectors using the del operator in an arbitrary coordinate system?

Question

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The image shows the full question, part (b) is what I'm interested in.

I'm stuck on how to use the del operator here to find out if $\vec{B}_0$, $\vec{E}_0$ and $\vec{q}$ are orthogonal. I tried using the cross prduct, but since there are no coordinate systems used in the definition of $\vec{B}$ and $\vec{E}$, it doesn't seem to give me anything useful.

Any advice? Thank you in advance.