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Math Help - intersection of vector functions problem

  1. #1
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    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!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ocryan View Post
    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?
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  3. #3
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    That does make sense, however, it seems that there must have been a miscalculation somewhere?

    These are my answer choices...
    -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?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ocryan View Post
    That does make sense, however, it seems that there must have been a miscalculation somewhere?

    These are my answer choices...
    -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).
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  5. #5
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    Ah, thank you very much. I appreciate your help.

    For my other question, z serves as the axis for the cylinder.
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ocryan View Post
    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).
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  7. #7
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    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.

    The answer choices are...

    (cost)i + (sint)j + (cos2t - 1)k
    (cost)i + (sint)j + (cos2t)k
    (cost)i + (sint)j + (1 - cos2t)k
    (sint)i + (cost)j + (cos2t - 1)k
    (sint)i + (cost)j + (cos2t)k
    (sint)i + (cost)j + (1 - cos2t)k


    I'm under the impression that since x(0) has to equal 1, then i must be cost, since cos(0) = 1. If that is correct that eliminates three of the six choices.
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