Tangent plane, parametric equation of the normal

I must calculate 2 tangent vectors (not parallel between themselves) to the surface in . Then find the tangent plane's to the surface in and I must find the parametric equation of the normal to the plane in the point .

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My attempt : I've done everything but I know I'm wrong and I misunderstand something.

A vector tangent to the surface in , .

I multiplied by in order to find . Here I realize that these 2 vectors must span the tangent plane to in so I could do the cross product to get the normal to the plane, but I did it another way.

. Which is the asked equation.

Here comes the big problem. According to my textbook (Calculus by Leithold), the implicit equation of the normal curve of the tangent plane in (1,2,4) can be found by the formula .

So I get .

Obviously this curve doesn't pass by . What happens?!

I put and wrote and in function of in order to get .

Now that I think I could take the gradient of and evaluate it in as I did to get .

I reach and this time the curve pass by . But why it doesn't work using Leithold's method?

Almost edit (I didn't post yet) I see my error! Actually I must not add the vector to . It works great!

Well, if it is not too much asked, are my answers correct? (For and I'm curious, especially for .)