Finding Intersection and Tangent Lines of Parametric Curves | Step-by-Step Guide

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Homework Help Overview

The discussion revolves around finding the point of intersection of the curves defined by the equations x^2 + y^2 = 1, z = 0 and the parametric equations x = cos(t), y = sin(t), z = t. Participants are also exploring how to determine the tangent lines to these curves at the intersection point.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to find the intersection point by substituting the parametric equations into the equation of the circle. They express uncertainty about the visualization of this process. Other participants discuss the expected uniqueness of the intersection point and inquire about the method for finding tangent lines to parametric curves in three-dimensional space.

Discussion Status

The discussion is active, with participants raising questions about the process of finding tangent lines and confirming the intersection point. Some guidance has been offered regarding differentiation as a method to find tangent lines, but no consensus has been reached on the overall approach.

Contextual Notes

Participants are navigating the complexities of visualizing intersections and tangents in three-dimensional space, with a focus on the implications of the parametric representation of curves.

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I need to find the point of intersection of the curves x^2 + y^2 =1, z= 0 and x=cost, y=sint, z=t. I plugged in the latter equation into the former and got (1,0,0) as an answer but I'm not exactly sure why that works, I can't visualize how plugging in the parts of a parametric equation will yield the point of intersection.

I also need to find the tangent lines to the curves at that point, and I'm not sure where to start.
 
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The two curves you have in the circle in the plane and a helix. You would expect that there is only one intersection point, and that will be at t=0.
 
Makes sense. How do I find the tangent line to a parametric curve in R3, though?
 
Differentiate with respect to t!
 

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