A normal line is perpendicular/orthogonal to a point on a surface, while a normal to a plane is perpendicular/orthogonal to a plane.
if we take partial derivatives of f(x,y) and evaluate it at a point we can get a tangent plane. Isn't the normal to this tangent plane equal to the normal line of the surface at that point?
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$\begingroup$The normal line is defined as the line that is perpendicular to the tangent line at the point of tangency.
unique line through P0 perpendicular to the plane.
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