Solving Tangent Plane and Perpendicular Plane at (1,1,1)

In summary, the conversation is discussing finding the equation of a tangent plane to a given surface, as well as finding an equation for a plane that is perpendicular to that tangent plane and passes through a given point. The participants are using the theorem that the gradient of a surface is normal to that surface to solve the problem. They also mention that there are an infinite number of planes that could fulfill the requirements, so they just need to pick one and set a convenient value for one variable.
  • #1
brads
2
0
Got a problem which should be easy but having trouble...

"Find the equation of the tangent plane to z=f(x,y)=x^2 + y^2 - 1 at point (1,1,1)
"Find an equation to a plane that is perpendicular to that tangent plane and also passes through the point (1,1,1)

Thanks!
 
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  • #2
Welcome to PF!

Are you familiar with the theorem that the gradient of a surface G(x,y,z)=0 is normal to that surface?
Use that hint (G(x,y,z)=z-f(x,y)).
 
  • #3
I figured out the first part of the problem the tangent plane to f(x,y) is 2x+2y-z-3=0. The normal vector of that plane is thus <2,2,-1>. I am lost on how to solve the next part, I keep trying different equations but coming up with too many variables.
 
  • #4
There's an infinity of planes which are normal to the tangent plane.
Pick one of them; the requirement is merely that some tangent vector in the tangent plane should be the normal to the plane you choose.
(the normal of the tangent plane is to be a tangent vector in the plane you choose)
 
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  • #5
"too many variables" only allows you to set one of them to any convenient value. Like arildno said, there are an infinite number of normal planes passing through a given point; so you get to pick anyone you like.
 

1. What is a tangent plane and a perpendicular plane?

A tangent plane is a flat surface that touches a curved surface at a specific point, without crossing or intersecting it. A perpendicular plane is a flat surface that intersects a given surface at a right angle.

2. Why is it important to solve for tangent plane and perpendicular plane at a specific point?

Solving for these planes at a specific point allows us to understand the behavior of a curved surface at that point. This information is useful in many fields such as physics, engineering, and mathematics.

3. How do you find the tangent plane and perpendicular plane at a given point?

To find the tangent plane, we use the partial derivatives of the given surface at the given point. To find the perpendicular plane, we use the normal vector of the given surface at the given point.

4. Can the tangent plane and perpendicular plane be the same at a given point?

No, the tangent plane and perpendicular plane are always different at a given point. The tangent plane is parallel to the surface at that point, while the perpendicular plane is perpendicular to the surface at that point.

5. How do you use the tangent plane and perpendicular plane at a given point in real-world applications?

The tangent plane and perpendicular plane can be used in various applications such as calculating the slope of a curved surface for designing roads or roller coasters, finding the optimal angle for mirrors in solar panels, and determining the direction of maximum curvature in 3D printing. They are also used in physics to understand the forces acting on a curved object at a specific point.

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