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

In summary, the conversation discusses finding the point of intersection of two curves, x^2 + y^2 = 1 and the parametric curve x=cost, y=sint, z=t. The point of intersection is (1,0,0) and it is found by plugging in the parametric equation into the first equation. The conversation also mentions the need to find tangent lines at that point, and the suggestion is to differentiate with respect to t.
  • #1
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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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  • #2
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.
 
  • #3
Makes sense. How do I find the tangent line to a parametric curve in R3, though?
 
  • #4
Differentiate with respect to t!
 

1. What is the intersection of curves?

The intersection of curves refers to the point(s) where two or more curves on a graph or surface intersect or overlap. It is the set of coordinates that satisfy both equations or functions of the curves.

2. How can the intersection of curves be determined?

The intersection of curves can be determined by graphing the curves and visually identifying the points where they intersect. Alternatively, algebraic methods such as substitution or elimination can be used to solve for the coordinates of the intersection point.

3. What does the intersection of curves represent in real-world applications?

The intersection of curves can represent various relationships in real-world applications such as the equilibrium point in economics, the solution to a system of equations in physics, or the point of intersection between two paths in navigation.

4. Can curves intersect at more than one point?

Yes, curves can intersect at more than one point. The number of intersection points depends on the number of curves and their equations. For example, two straight lines can intersect at one point, while a line and a parabola can intersect at two points.

5. What happens if there is no intersection of curves?

If there is no intersection of curves, it means that the curves do not overlap or intersect at any point. This could indicate that the equations or functions are not compatible or that the curves are parallel or do not intersect in the given domain.

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