Problem:

Find an equation of the tangent plane to the ellipsoid x^2+4y^2+z^2=18 at the point (1,2,1) and determine the acute angle that this plane makes with the xy-plane.

Work:

Equation of Tangent Plane is: 2(x-1)+16(y-2)+2(z-1)

Now, this is how I tried to find the acute angle between this plane and the xy-plane:

I used the equation cosθ= n1·n2/|n1||n2|, where n1 and n2 are the normal vector (one of the tangent plane and one of the xy plane).

The normal vector I used for the xy plane was <0,0,1> and the normal vector for the tangent plane is <2,16,2>

Plugging everything into the equation I found that the acute angle is 1.44739 radians or 82.929 degrees.

Can someone verify that this technique is correct? Mainly, I think I need to know if that was the correct normal vector to use for the xy-plane, and if the method I used in general is correct.

Thanks!