Quickie: vector normal to surface

In summary, the conversation discusses finding the vector normal to z = x^2 + y^2 - 3 at a given point. The suggestion is made to view the surface as a level surface of some function, and the concept of computing the magnitude of a vector is discussed. The error is pointed out and the conversation ends with the question being answered.
  • #1
ilyas.h
60
0

Homework Statement


Find vector normal to z = x^2 + y^2 - 3 at point r = (2, -1, 2)

Homework Equations

The Attempt at a Solution


normal.png


here is the markscheme. I understand how to find the gradient, but i don't understand how they calculated the magnitude.

thanks
 
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  • #2
## z = x^2 +y^2 - 3 ## is satisfied by points that form a three dimensional surface, and it's helpful to view that surface as the level surface of some function. How about viewing it as the level surface ## f(x,y,z) = 0 ## for ## f(x,y,z) = z - x^2 - y^2 + 3 ##? Does that clarify things?
 
  • #3
Geofleur said:
## z = x^2 +y^2 - 3 ## is satisfied by points that form a three dimensional surface, and it's helpful to view that surface as the level surface of some function. How about viewing it as the level surface ## f(x,y,z) = 0 ## for ## f(x,y,z) = z - x^2 - y^2 + 3 ##? Does that clarify things?

nope. Take the magnitude of the llevel surface?
 
  • #4
ilyas.h said:
nope. Take the magnitude of the llevel surface?
What's the magnitude of the vector -4i + 2j + k ? This is the gradient vector at (2, -1, 2)
 
  • #5
SteamKing said:
What's the magnitude of the vector -4i + 2j + k ? This is the gradient vector at (2, -1, 2)

how does that help? i need to know how to compute the magnitude.
 
  • #6
ilyas.h said:
how does that help? i need to know how to compute the magnitude.
Good Lord, you're working gradient problems, but you've skipped basic vector arithmetic.

If V = ai + bj + ck, then |V| = (a2 + b2 + c2) 1/2

Also |V| = (VV)1/2, where ⋅ signifies the dot product of two vectors.
 
  • #7
SteamKing said:
Good Lord, you're working gradient problems, but you've skipped basic vector arithmetic.

If V = ai + bj + ck, then |V| = (a2 + b2 + c2) 1/2

Also |V| = (VV)1/2, where ⋅ signifies the dot product of two vectors.

My fault all along. I was trying to calculate the magnitude of the surface instead of the magnitude of the gradient of the surface.

Yes i know how to calculate the magnitude of a vector thank you very much.

answered.
 

What is a vector normal to surface?

A vector normal to surface is a vector that is perpendicular to the surface at a specific point. It is also known as a surface normal or a unit normal vector.

Why is a vector normal to surface important?

A vector normal to surface is important because it helps determine the orientation and direction of a surface. It is also used in many mathematical and physical applications, such as calculating surface area and determining the direction of light reflection.

How is a vector normal to surface calculated?

A vector normal to surface is calculated by taking the cross product of two tangent vectors on the surface at a specific point. The result is a vector that is perpendicular to both tangent vectors and therefore normal to the surface.

What is the difference between a unit normal vector and a regular normal vector?

A unit normal vector has a magnitude of 1, while a regular normal vector can have any magnitude. In other words, a unit normal vector is a normalized version of a regular normal vector, meaning it has the same direction but a different magnitude.

How is a vector normal to surface used in computer graphics?

In computer graphics, a vector normal to surface is used to calculate lighting and shading effects on 3D objects. It is also used in computer animation to determine the direction of surface movement and to create realistic textures.

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