The z-component of your cross-product is incorrect:
That should prove much simpler, particularly when you take the norm.
Find the parametric representation of the ellipsoid using the Euler's angles. Then calculate the normal vector and the first quatratic form at (1,1,1).
Solution so far:
I use the method I learned in class and got
Now, the normal vector
But I found that , in which is getting a bit ridiculous, am I doing the right thing?