To address this problem, let's go through the steps to find the Lagrangian and the angle from the conditions provided.

The function $F(x, y, z)$ is given by:

$F(x, y, z) = x^2 + \frac{3}{2}y^2 + \frac{1}{3}z^2 + x y z + 2x + \frac{2}{3}y + 2$

The constraint given by the surface is:

$x^2 + y^2 + z^2 = 2, \quad x \geq 0, \quad y \geq 0, \quad z \geq 0$

To solve this using the Lagrange multipliers method, follow these steps:

1. Define the Lagrangian Function:

   The Lagrangian function $\mathcal{L}(x, y, z, \lambda)$ incorporates both the objective function (