orthogonal transformations and rigid motions?

i am asking about 2 main things...

how would one go about proving that the vector product is invariant under an orthogonal transformation?

also, how would one go about showing that the arc length, curvature, and torsion of a parametrized curve are invariant under rigid motions?

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for the first one, i had the idea that you could consider the vector product of Ax with Ax (A an orthogonal transformation)...and then just take a basis of A, say v1, v2, v3, where v1, v2, v3 are just rows of A. then you could express the vector product as a determinant of a matrix whose entries are just inner products...then i suppose it might fall out naturally...

but for the second one, clearly the arclength is invariant wrt translation and orthogonal transformations, but how can i express that mathematically? is it the same as considering invariance of the norm of the tangent vector over translation and orthogonal transformations? it seems correct, but almost too easy...