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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.

Identify the surface in R^3: 1. The surface r = 3 sec(θ) is a plane, and the equation in rectangular coordinates is x = 3. 2. The surface √3 z = √(x^2 + y^2) is a cone, and the equation in spherical coordinates is φ = π/6.

This problem involves identifying surfaces in R^3 using cylindrical and spherical coordinates. It covers the equation of a plane r = 3 sec(θ) and a cone √3 z = √(x^2 + y^2), with their conversions into rectangular and spherical coordinates.

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