Equation of the normal plane to the surface at (1,0,1)

[petrol.veem]

Find an equation of the normal plane to the surface at the point (1,0,1)

x^2 + y^2 - z^2 - 2xy + 4xz = 4

I know how to find equations for tangent planes, but how does one go about finding equations for normal planes?

 

[pka]

First you need to finf the gradiant:
\(\displaystyle \begin{array}{l}
F = x^2 + y^2 - z^2 - 2xy + 4xz \\
\nabla F = F_x i + F_y j + F_z k \\
\end{array}.\)

The normal of the plane is \(\displaystyle \nabla F\left( {1,0,1} \right).\)

 

[petrol.veem]

pka, do you mean the normal to the tangent plane is given by the gradient vector?

the normal plane is something different than the normal vector, is it not? the normal plane should be parallel to the normal vector of the tangent plane.

 

[pka]

Sorry, I miss-read the question. I of course thought of a tangent plane.
There are two classical textbooks on Vector Analysis, one by Davis and Snider and the other by Barr. I have taught from both and neither discusses a normal plane to a surface. Nor do I ever remember seeing the idea before. Normal planes to curves yes, but to surfaces? I cannot visualize what that would be.

 

[galactus]

I think a tangent plane and normal line is what is meant.

 

[pka]

I think a tangent plane and normal line is what is meant.

Now that would make sense!
Why did I not think of that?

 

[petrol.veem]

the question i am dealing with says specifically normal plane to the surface. no typo.

the second part of the question (which i didn't write down) goes on to ask about the normal line to the surface as well.

i am confused.

 

[pka]

the question i am dealing with says specifically normal plane to the surface. no typo.

the second part of the question (which i didn't write down) goes on to ask about the normal line to the surface as well.

i am confused.

No one will blame you for that!
Please know that we all make mistakes. Every textbook has mistakes!