Tangent vector to curve of intersection of 2 surfaces

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Homework Help Overview

The problem involves finding the tangent vector at a specific point to the curve formed by the intersection of two surfaces defined by the equations z = x² + y² and z = x + y.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods to approach the problem, including parametrization of the curve and utilizing the tangent planes of the surfaces. There is also a suggestion to find the gradients of the surfaces and consider their cross product.

Discussion Status

The discussion is active, with participants exploring different methods and confirming that the gradients of the surfaces can be used to find the tangent vector. There is an acknowledgment that only the direction of the tangent vector is necessary.

Contextual Notes

Participants note the importance of ensuring that the gradients are not parallel, which could affect the validity of the approach being discussed.

plexus0208
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Homework Statement


Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y.

Homework Equations



The Attempt at a Solution


I haven't started the problem, because I'm not sure what the first thing to do is.
Do I have to parametrize the equations first? Then what?
 
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there's a couple of ways to do it... you could do it by finding a parametrisation of the curve..

an easier way woud be to notice that the tangent vector to the curve, will lie in both the tangent planes of the surfaces, then go form there
 
Oh!
If I find the gradient of each surface, and then cross the two gradients, that's it right?
 
yes, as the gradients will be normal to the tangent planes, and hopefully they are not parallel...

note that only the direction of the tangent vector is required in this case
 

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