Let's go over the details in the image.

The question asks to identify surfaces in $\mathbb{R}^3$ for the following cases:

1. The surface $r = 3 \sec(\theta)$ is identified as a plane, and the equation in rectangular coordinates is given as $x = 3$.

   • Explanation: In cylindrical coordinates, $r$ represents the radial distance, and $\theta$ the angle. The equation $r = 3 \sec(\theta)$ translates to $x = 3$ in rectangular coordinates, which is a vertical plane parallel to the $yz$-plane.

2. The surface $\sqrt{3} z = \sqrt{x^2 + y^2}$ is identified as a cone, and the equation in spherical coordinates is given as $\phi = \frac{\pi}{6}$.

   • Explanation: This equation represents a cone. In cylindrical coordinates, $\sqrt{x^2 + y^2} = r$, and $z = \frac{r}{\sqrt{3}}$. In spherical coordinates, the angle $\phi$ is the angle between the cone's surface and the $z$-axis, which equals $\frac{\pi}{6}$, matching the typical equation for a cone.

Would you like more detailed explanations or further clarifications on this?

Here are five related questions you might consider:

1. How do cylindrical and spherical coordinates relate to each other geometrically?
2. How can you convert between rectangular and spherical coordinates?
3. Why does the equation $r = 3 \sec(\theta)$ represent a vertical plane?
4. What other surfaces can be described in spherical coordinates?
5. How do we generalize cones in higher-dimensional spaces?

Tip: When switching between coordinate systems, visualize how each system defines points in space—this helps to understand the shape and surface described by the equations.