That problem doesn't make much sense to me as stated. Have you included the full statement of the problem?

There are numerous "circular cylinder of radius 1" in 3-space. Is that freedom of positioning such a circlar cylinder what this problem is about?

That reads that the "vector function" you're asked to find is a vector form for a curve.

But the meaning of "the curve of intersection is x(0)=0" is very unclear. Is it the constant curve at the origin? Is it *some* curve satisfying x = 0 at time 0? That isn't even a point in 3 space. If it's *some* curve, then why the word "the". Saying "x(0)=0" is simply not describing "THE curve of intersection" for those two objects in 3 space.

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What is this circular cylinder of radius 1? It seems to me that both of the below satisfy "the curve of intersection is x(0)=0". I really just don't get it.