# Finding normal vector to a surface

• Kuma
In summary, to find a normal vector to a parameterized surface, you can take the cross product of the tangent vectors, Tu and Tv, which are defined as (dx/du, dy/du, dz/du) and (dx/dv, dy/dv, dz/dv), respectively. It does not matter which points of u and v you use to calculate the cross product. However, if the surface is curved, the cross product should not be equal to zero. If the surface is a plane, the cross product of the tangents will be equal to zero and thus cannot be used to find the normal vector.
Kuma

## Homework Statement

Given a parameterized surface:

C(u,v) = (3 cos u sin v, 2 sin u sin v, cos v) 0<u<2pi, 0<v<pi

I have to find a normal vector to that surface.

## The Attempt at a Solution

So tangent vectors can be Tu = (dx/du, dy/du, dz/du) and Tv = (dx/dv, dy/dv, dz/dv)

And I can take the cross of those to find a normal vector. But what points of u and v do i use? The cross product gave me:

(-2sin^2 v cos u, 3sin^2 v sin u, 6sin^2 u sin^2 v - 6 cos^2 u sin^2 v)

But what points of u and v do i use?
What difference does it make? (And why?)

It shouldn't make a difference. Do i just plug in the endpoints of u and v? ie would (0,0) work? I get (0,0,0) if I use that point. Not a vector...

Why shouldn't it make a difference?
What happens if the surface is curved?

Right. When the surface is curved the cross product of the tangents shouldn't be 0.

You are saying that the cross product of the tangents to a plane (flat) surface are zero?
Then how would you find the normal vector to a plane surface?

## 1. What is a normal vector?

A normal vector is a vector that is perpendicular, or at a right angle, to a surface at a specific point. It is used to indicate the direction in which the surface is facing.

## 2. Why is it important to find the normal vector to a surface?

Finding the normal vector to a surface is important because it provides crucial information about the surface, such as its orientation and direction. It is also necessary for many mathematical calculations and physical simulations.

## 3. How do you calculate the normal vector to a surface?

The normal vector to a surface can be calculated using the partial derivatives of the surface's equation. Specifically, the normal vector is equal to the cross product of the partial derivatives of the surface with respect to its parameters.

## 4. Can the normal vector change at different points on a surface?

Yes, the normal vector can change at different points on a surface. This is because the direction and orientation of the surface can vary, and the normal vector is dependent on these factors.

## 5. What are some real-world applications of finding the normal vector to a surface?

Finding the normal vector to a surface has many practical applications in fields such as engineering, physics, and computer graphics. It is used in designing and analyzing structures, calculating forces and moments, and creating realistic 3D models of surfaces.

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