To convert a point from rectangular coordinates $(x, y, z)$ to cylindrical coordinates $(r, \theta, z)$, we use the following formulas:

1. Radial distance $r = \sqrt{x^2 + y^2}$
2. Angle $\theta = \tan^{-1}\left(\frac{y}{x}\right)$
3. The $z$-coordinate remains the same.

Given the point $(6, 2\sqrt{3}, -9)$:

Step 1: Calculate $r$

$r = \sqrt{x^2 + y^2} = \sqrt{6^2 + (2\sqrt{3})^2}$ $= \sqrt{36 + 4 \cdot 3} = \sqrt{36 + 12} = \sqrt{48} = 4\sqrt{3}$

Step 2: Calculate $\theta$

$\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{2\sqrt{3}}{6}\right)$ $= \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}$

Step 3: Retain the $z$-coordinate

The $z$-coordinate remains the same, so $z = -9$.

Final Answer

The cylindrical coordinates are: $(r, \theta, z) = \left(4\sqrt{3}, \frac{\pi}{6}, -9\right)$

Would you like further details or have any questions?

1. How do you convert from cylindrical to rectangular coordinates?
2. What are cylindrical coordinates used for in real-world applications?
3. How do you handle negative $z$-values in cylindrical coordinates?
4. Can cylindrical coordinates be converted to spherical coordinates?
5. How would the process change if given points were in 3D space with spherical coordinates?

Tip: Always check if the angle $\theta$ needs to be adjusted based on the quadrant of $(x, y)$.