Tangent/Normal planes, intersections, vector tangent

In summary, the conversation involves finding the normal and tangent plane to a given surface at a specific point, as well as a vector tangential to the curve of intersection of the surface and a given plane at the same point. The method used is to find the normal and tangent plane equations using the given point and the surface's partial derivatives, and then find the vector by crossing the normals of the two surfaces.
  • #1
Gameowner
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Homework Statement



Surface (s) given by

x^3*+5*y^2*z-z^2+x*y=0

question ask to find normal and tangent plane to S at the point P=(1,-1,0) then find vector tangential to curve of intersection of S and the plane x+z=1 at point P.

Homework Equations


The Attempt at a Solution



So I started off finding the normal first using n=(fx,fy,-1) which gave me n=(-2,-1,1).

Then proceeding to the tangent plane using the tangent plane equation fx(x-x0)+fy(y-y0)-(z-z0)=0, which gave me the tangent plane 2*x+y-z-1=0.

Please correct me if my method is right or wrong for the above.

But my real question is the last part of the problem statement, which is to find the vector tangential to the curve intersecting x+z=1.

My guess is to find the gradient of the tangent vector and the gradient of the given curve(x+z=1) and take the cross product of the 2? I have no idea where to begin the last, part, any hint or help would be much appreciated.
 
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  • #2
Gameowner said:

Homework Statement



Surface (s) given by

x^3*+5*y^2*z-z^2+x*y=0

question ask to find normal and tangent plane to S at the point P=(1,-1,0) then find vector tangential to curve of intersection of S and the plane x+z=1 at point P.



Homework Equations





The Attempt at a Solution



So I started off finding the normal first using n=(fx,fy,-1) which gave me n=(-2,-1,1).


That formula for the normal is for when the equation is in the form z = f(x,y). Your equation is not in that form but in the implicit form f(x,y,z)=0. Use [itex]\nabla f[/itex].

Then proceeding to the tangent plane using the tangent plane equation fx(x-x0)+fy(y-y0)-(z-z0)=0, which gave me the tangent plane 2*x+y-z-1=0.

Please correct me if my method is right or wrong for the above.

But my real question is the last part of the problem statement, which is to find the vector tangential to the curve intersecting x+z=1.

My guess is to find the gradient of the tangent vector and the gradient of the given curve(x+z=1) and take the cross product of the 2? I have no idea where to begin the last, part, any hint or help would be much appreciated.

The "gradient of the tangent vector" doesn't make any sense. Just cross the normals to the two surfaces.
 

1. What is a tangent plane?

A tangent plane is a flat surface that touches a curved surface at only one point. It is perpendicular to the normal line at that point and represents the best approximation of the curved surface at that point.

2. How is a normal plane different from a tangent plane?

A normal plane is also a flat surface that touches a curved surface at only one point. However, it is perpendicular to the tangent line at that point and represents the best approximation of the curved surface's direction at that point.

3. What does the vector tangent represent?

The vector tangent represents the direction of the tangent line at a specific point on a curved surface. It is a vector that is tangent to the surface at that point and points in the direction of the surface's local orientation.

4. How do you find the intersection of two tangent planes?

The intersection of two tangent planes can be found by setting their equations equal to each other and solving for the variables. This will give you the coordinates of the point where the two planes intersect.

5. Can a tangent plane ever be parallel to a normal plane?

No, a tangent plane and a normal plane can never be parallel. This is because they are always perpendicular to each other at the point of tangency on a curved surface.

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