Check if I did this problem correctly please

[penguinnnnnx5]

Homework Statement

Check that the point (-1,1,2) lies on the given surface. Then find a vector normal to the surface and an equation for the tangent plane to the surface at (-1,1,2).

z=x² + y²

Homework Equations

The Attempt at a Solution

(2) = (-1)² + (1)²

2 = 2; (-1, 1, 2) lies on the surface z=x² + y²

----------

0 = -z + x² + y²
f(x,y,z) = -z + x² + y²

∇f = <2x, 2y, -1>
∇f (-1,1,2) = <-2,2,-1>

tangent plane: -2(x+1) + 2(y-1) - (z-2) = 0

-2x + -2 + 2y - 2 - z + 2 = 0

-2x + 2y - z = 2 <------- Tangent plane equation

normal line: < -2, 2, -1>

as unit vector: (1/√3) <-2, 2, -1>

Did I do this correctly?

 

[brmath]

It looks right to me.