1. (a) (10 pts) Find the spherical coordinates of $(\sqrt{3},-3,2)$.
(b) (10 pts) Find the cylindrical coordinates of $(-\sqrt{2},-2,2 \sqrt{3})$.

Step 1 ：Given the Cartesian coordinates $(\sqrt{3},-3,2)$, we need to convert these to spherical coordinates.

Step 2 ：The spherical coordinates are given by $(r, \theta, \phi)$ where $r$ is the distance from the origin to the point, $\theta$ is the angle in the xy-plane from the positive x-axis to the line segment connecting the origin and the projection of the point onto the xy-plane, and $\phi$ is the angle from the positive z-axis to the line segment connecting the origin and the point.

Step 3 ：We use the following formulas for conversion: $r = \sqrt{x^2 + y^2 + z^2}$, $\theta = \arctan(\frac{y}{x})$, and $\phi = \arccos(\frac{z}{r})$.

Step 4 ：Substitute the given Cartesian coordinates into these formulas: $x = 1.7320508075688772$, $y = -3$, $z = 2$.

Step 5 ：Calculate $r = \sqrt{x^2 + y^2 + z^2} = 4.0$.

Step 6 ：Calculate $\theta = \arctan(\frac{y}{x}) = -1.0471975511965979$.

Step 7 ：Calculate $\phi = \arccos(\frac{z}{r}) = 1.0471975511965979$.

Step 8 ：Final Answer: The spherical coordinates of $(\sqrt{3},-3,2)$ are \(\boxed{(4, -1.047, 1.047)}\).