application of partial derivatives tangent line and normal plane

Question

please explain did normal line is parallel to the tangent plane of a surface or It intersects with the tangent plane of a surface or
It is contained in the tangent plane of a surface

Answer 1

the normal to the surface in a point coincides with the normal to the tangent plane at that point

where the equantion for the tangent plane is:

$$z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$

thus the normal at the point $(x_0,y_0,z_0)$ is:

$$\vec{n}=(f_x(x_0,y_0),f_y(x_0,y_0),-1)$$