text: # Thread: intersection of vector functions problem
  
  1. ## 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!
  
  2. 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?
  
  3. 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?
  
  4. 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).
  
  5. Ah, thank you very much. I appreciate your help.
  
  For my other question, z serves as the axis for the cylinder.
  
  6. 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).
  
  7. 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.
explanation: The document is a discussion thread where a user asks for help with a math problem involving the intersection of a sphere and a plane, and later asks about the vector function for a circular cylinder. The thread includes the user's question and a detailed response from another user providing the solution and additional context.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Thread: intersection of vector functions problem
  
  1. ## 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!
  
  2. 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?
  
  3. 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?
  
  4. 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).
  
  5. Ah, thank you very much. I appreciate your help.
  
  For my other question, z serves as the axis for the cylinder.
  
  6. 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).
  
  7. 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread where a user asks for help with a math problem involving the intersection of a sphere and a plane, and later asks about the vector function for a circular cylinder. The thread includes the user's question and a detailed response from another user providing the solution and additional context.
  
  Label: question-and-answer