# Quickie: vector normal to surface

1. Oct 16, 2015

### ilyas.h

1. The problem statement, all variables and given/known data
Find vector normal to z = x^2 + y^2 - 3 at point r = (2, -1, 2)

2. Relevant equations

3. The attempt at a solution

here is the markscheme. I understand how to find the gradient, but i dont understand how they calculated the magnitude.

thanks

2. Oct 16, 2015

### Geofleur

$z = x^2 +y^2 - 3$ 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. Oct 16, 2015

### ilyas.h

nope. Take the magnitude of the llevel surface?

4. Oct 16, 2015

### SteamKing

Staff Emeritus
What's the magnitude of the vector -4i + 2j + k ? This is the gradient vector at (2, -1, 2)

5. Oct 16, 2015

### ilyas.h

how does that help? i need to know how to compute the magnitude.

6. Oct 16, 2015

### SteamKing

Staff Emeritus
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. Oct 16, 2015

### ilyas.h

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.