1. **Problem Statement:** (a) Convert points A(2,2,1) and B(5,5,4) from Cartesian $(x,y,z)$ to cylindrical coordinates $(r,\theta,z)$ where $r=\sqrt{x^2+y^2}$ and $\theta=\tan^{-1}(y/x)$. 2. **Formula and Rules:** - Radial component: $r=\sqrt{x^2+y^2}$ - Angular component: $\theta=\tan^{-1}(\frac{y}{x})$ - Vertical component: $z=z$ - Angle $\theta$ is in radians, measured from positive x-axis. 3. **Convert Point A:** $$r_A=\sqrt{2^2+2^2}=\sqrt{8}=2\sqrt{2}$$ $$\theta_A=\tan^{-1}(\frac{2}{2})=\tan^{-1}(1)=\frac{\pi}{4}$$ $$z_A=1$$ 4. **Convert Point B:** $$r_B=\sqrt{5^2+5^2}=\sqrt{50}=5\sqrt{2}$$ $$\theta_B=\tan^{-1}(\frac{5}{5})=\tan^{-1}(1)=\frac{\pi}{4}$$ $$z_B=4$$ 5. **Interpretation Differences:** - Cartesian coordinates describe position by perpendicular distances along x, y, and z axes. - Cylindrical coordinates describe position by distance from z-axis (radial), angle around z-axis (angular), and height (vertical). - Direction in Cartesian is vector difference in x,y,z; in cylindrical, direction involves changes in $r$, $\theta$, and $z$. 6. **Q3 (Intersection of Cylinder and Plane):** - Cylinder: $r=4$, Plane: $z=3$ - Parametric points on intersection: choose $\theta=0, \frac{\pi}{2}, \pi$ - Points in cylindrical: $(4,0,3), (4,\frac{\pi}{2},3), (4,\pi,3)$ - Convert to Cartesian: $$x=4\cos(0)=4, y=4\sin(0)=0, z=3$$ $$x=4\cos(\frac{\pi}{2})=0, y=4\sin(\frac{\pi}{2})=4, z=3$$ $$x=4\cos(\pi)=-4, y=4\sin(\pi)=0, z=3$$ 7. **Q4 (Cylindrical Region Corners):** - Corners at combinations of $r=0,6$, $\theta=0,\frac{\pi}{2}$, $z=0,4$ - Compute Cartesian for each: For example, $(r,\theta,z)=(6,0,0)$: $$x=6\cos(0)=6, y=6\sin(0)=0, z=0$$ - Similarly for all 8 corners. 8. **Q5 (Spherical Corridor Corners):** - Corners at $\rho=5,8$, $\phi=0,\frac{\pi}{3}$, $\theta=0,\frac{\pi}{2}$ - Cartesian: $$x=\rho \sin\phi \cos\theta, y=\rho \sin\phi \sin\theta, z=\rho \cos\phi$$ - Compute for each combination. 9. **Q6 (VR Camera Points):** - Given $\rho=1$, compute points for $\theta=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ and $\phi=\frac{\pi}{6}, \frac{\pi}{4}$ - Use spherical to Cartesian: $$x=\sin\phi \cos\theta, y=\sin\phi \sin\theta, z=\cos\phi$$ **Final answers:** (a) Cylindrical coordinates: - A: $(2\sqrt{2}, \frac{\pi}{4}, 1)$ - B: $(5\sqrt{2}, \frac{\pi}{4}, 4)$ (b) Interpretation explained above. (c) Visualization code provided by user. (d) Intersection points: - $(4,0,3)$, $(0,4,3)$, $(-4,0,3)$ in Cartesian. (e) Cylindrical region corners (example): - $(0,0,0)$, $(6,0,0)$, $(0,6,0)$, $(6,6,0)$, $(0,0,4)$, $(6,0,4)$, $(0,6,4)$, $(6,6,4)$ in Cartesian after conversion. (f) Spherical corridor corners computed similarly. (g) VR camera points computed as above.