1. **Problem Statement:** Convert points A(2, 2, 1) and B(5, 5, 4) from Cartesian to cylindrical coordinates and analyze their components. 2. **Formula for Cylindrical Coordinates:** Cylindrical coordinates $(r, \theta, z)$ relate to Cartesian $(x, y, z)$ as: $$r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right), \quad z = z$$ 3. **Convert Point A:** $$r_A = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$$ $$\theta_A = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4}$$ $$z_A = 1$$ So, $A = \left(2\sqrt{2}, \frac{\pi}{4}, 1\right)$. 4. **Convert Point B:** $$r_B = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$$ $$\theta_B = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1) = \frac{\pi}{4}$$ $$z_B = 4$$ So, $B = \left(5\sqrt{2}, \frac{\pi}{4}, 4\right)$. 5. **Interpretation of Components:** - Radial component $r$ is the distance from the $z$-axis. - Angular component $\theta$ is the angle in the $xy$-plane from the positive $x$-axis. - Vertical component $z$ is the height, same as Cartesian $z$. 6. **Difference in Interpretation:** - In Cartesian, position is given by $(x,y,z)$ coordinates along perpendicular axes. - In cylindrical, position is given by distance from axis ($r$), angle around axis ($\theta$), and height ($z$). - Direction of motion in Cartesian is vector difference $(\Delta x, \Delta y, \Delta z)$. - In cylindrical, direction involves changes in $r$, $\theta$, and $z$, which can be more intuitive for rotational or radial motion. 7. **Desmos Function:** The drone path is a straight line between points $A$ and $B$: $$y = \frac{5-2}{5-2}(x-2) + 2 = x$$ But since it's 3D, parametric form is better: $$x = 2 + 3t, \quad y = 2 + 3t, \quad z = 1 + 3t, \quad t \in [0,1]$$ Final answers: - Point A cylindrical: $\left(2\sqrt{2}, \frac{\pi}{4}, 1\right)$ - Point B cylindrical: $\left(5\sqrt{2}, \frac{\pi}{4}, 4\right)$