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YES you are correct
Yes, you are right, because cross product vector is always perpendicular to both of the vectors we are multiplying, thus it's perpendicular to vector v. Dot product has in itself the cosine of the angle between two given vectors, and we know that the cosine of 90 is 0.
I think this concept can be applied to div(curl F) = 0 as well
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It's also true because:
a dot (b cross c) = (a cross b) dot c
Btw. If you didn't know:
a dot (b cross c) = c dot (a cross b) = b dot (c cross a)
also and from the fact that dot-product is commutative and the second permutation we get the fact that one can exchange the dot and cross in triple product.
These are very easy to prove using the matrix presentation of the triple product.