# intersection of vector functions problem

• October 9th 2009, 04:52 PM
ocryan
intersection of vector functions problem
Hello everyone,

I am stuck on this question, although I suspect it is easy enough to solve, my textbook isn't providing a lot of help, if y'all could explain how to solve this I'd appreciate it!

Find a vector function whose graph is the curve of intersection of the sphere x^2 + y^2 + z^2 = 10 and the plane x = -1.

Thanks!
• October 9th 2009, 05:08 PM
Chris L T521
Quote:

Originally Posted by ocryan
Hello everyone,

I am stuck on this question, although I suspect it is easy enough to solve, my textbook isn't providing a lot of help, if y'all could explain how to solve this I'd appreciate it!

Find a vector function whose graph is the curve of intersection of the sphere x^2 + y^2 + z^2 = 10 and the plane x = -1.

Thanks!

Note that when x=-1, $x^2+y^2+z^2=10\implies y^2+z^2=9$.

So the vector valued function $\mathbf{f}(x(t),y(t),z(t))=\left<-1,3\cos t,3\sin t\right>$ should work (to get the last two components, parameterize the equation for the circle $y^2+z^2=9$).

Does this make sense?
• October 9th 2009, 05:18 PM
ocryan
That does make sense, however, it seems that there must have been a miscalculation somewhere?

-3 sin t, 3 cos t, -1
3 sin t, 1 , -3 sin t
1, 3 cos t, -3 sin t
3 cos t, -3 sin t, 1
-1, 3 sin t, -3 cos t
3 cos t, -1, -3 sin t
plane and sphere don't intersect.

Given those, your answer of -1, 3 cos t, 3 sin t, isn't available...

Oh, also a question I had that is a part of another question, what is the vector function for a circular cylinder of radius 1?
• October 9th 2009, 05:25 PM
Chris L T521
Quote:

Originally Posted by ocryan
That does make sense, however, it seems that there must have been a miscalculation somewhere?

-3 sin t, 3 cos t, -1
3 sin t, 1 , -3 sin t
1, 3 cos t, -3 sin t
3 cos t, -3 sin t, 1
-1, 3 sin t, -3 cos t
3 cos t, -1, -3 sin t
plane and sphere don't intersect.

Given those, your answer of -1, 3 cos t, 3 sin t, isn't available...

Oh, also a question I had that is a part of another question, what is the vector function for a circular cylinder of radius 1?

In this case, $\left<-1,3\sin t,-3\cos t\right>$ works as well since $y^2+z^2=\left(3\sin t\right)^2+\left(-3\cos t\right)^2=9\left(\sin^2t+\cos^2t\right)=9$ which still gives us the equation of the circle we started with.

To answer the second question, I need to know which axis acts as the center of each circular cross section since I can come up with 3 different equations (one for each coordinate axis).
• October 9th 2009, 05:31 PM
ocryan
Ah, thank you very much. I appreciate your help.

For my other question, z serves as the axis for the cylinder.
• October 9th 2009, 05:40 PM
Chris L T521
Quote:

Originally Posted by ocryan
Ah, thank you very much. I appreciate your help.

For my other question, z serves as the axis for the cylinder.

In this case, z can be anything, so lets say $z=t$. Then we just have the equation for a circle $x^2+y^2=1$. This is parameterized by $x=3\cos t$ and $y=3\sin t$.

Therefore, the parameterization for the cylinder would be $\left<3\cos t,3\sin t,t\right>$.

Does this make sense?

(If this is another multiple choice question, note that the first two elements can be swapped [if needed] and can be postive,negative, or any combination thereof).
• October 9th 2009, 05:48 PM
ocryan
Hmm, maybe I should post the full question.

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

Which vector function has the curve of intersection as its graph if x(0) = 1.