Let the surface be defined as $F(x, y, z) = x^2 + y^2 - z - 1 = 0$.
The normal vector is given by the gradient:
$\nabla F = 2x \hat{i} + 2y \hat{j} - \hat{k}$
At the point $(1, 1, 1)$:
$\nabla F = 2(1)\hat{i} + 2(1)\hat{j} - \hat{k} = 2\hat{i} + 2\hat{j} - \hat{k}$
Magnitude of the normal vector:
$|\nabla F| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$
Unit normal vector:
$\hat{n} = \frac{\nabla F}{|\nabla F|} = \frac{2\hat{i} + 2\hat{j} - \hat{k}}{3}$