Thread: Finding a Vector Function

Hey guys,
I am having a bit of a problem with vector functions. This problem says: A circular cylinder of radius 1 and the parabolic cylinder z=2x^2 intersect. What is the vector function if the curve of intersection is x(0)=0?

I first set x = cos(t) and y = sin(t); then solved for z = 2 cos^2(t). Would this be correct? Or am I approaching this problem incorrectly?

Any feed and help is appreciated.

Thanks

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

Originally Posted by Beevo

A circular cylinder of radius 1 and the parabolic cylinder z=2x^2 intersect.

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?

Originally Posted by Beevo

What is the vector function if the curve of intersection is x(0)=0?

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.

Yes, that looks good to me. The parabolic cylinder and the circular cylinder intersect in a curve given by x= cos(t), y= sin(t),

Well the problem provided a graph. First, the graph of the cylinder has the z-axis as the central axis, so I am assuming the equation would be x^2 + y^2 = 1. Then the other equation is z=2x^2, which intersects with the cylinder. Then the main question states, which vector function has the curve of intersection as its graph if x(0)=0.

Also, does the x(0)=0 matter? What if it was x(0)=1.

It is x(0) = 1 if x(t) = cos(t).
Although it's not how to usually do it, in this case, if that's basically the form you're seeking, make it x(t) = sin(t), y(t) = cos(t) in order to have x(0) = 0.
Though again, who knows what this problem is asking, so who knows if that's a reasonable answer. Surely not me.

Edit:

Originally Posted by Beevo

Well the problem provided a graph. First, the graph of the cylinder has the z-axis as the central axis, so I am assuming the equation would be x^2 + y^2 = 1.

Yes - that's right. That makes the problem much more clear.

Originally Posted by johnsomeone

It is x(0) = 1 if x(t) = cos(t).
Although it's not how to usually do it, in this case, if that's basically the form you're seeking, make it x(t) = sin(t), y(t) = cos(t) in order to have x(0) = 0.
Though again, who knows what this problem is asking, so who knows if that's a reasonable answer. Surely not me.

Edit:

Yes - that's right. That makes the problem much more clear.

Yeah that is it. Thanks for your help.