Evaluate the surfaces in ℝ3 that are represented by the following equations. i. y = 2 ii. z = 6

In \( \mathbb{R}^3 \), the equations \( y = 2 \) and \( z = 6 \) represent surfaces. Let's evaluate what these surfaces look like.

### i. Surface represented by \( y = 2 \)

The equation \( y = 2 \) describes a **plane** in \( \mathbb{R}^3 \). Specifically:
- This plane is parallel to the \( xz \)-plane because it doesn't involve the \( x \) or \( z \) coordinates.
- The value of \( y \) is fixed at 2, so every point on this plane has the form \( (x, 2, z) \), where \( x \) and \( z \) can be any real numbers.

Thus, the surface is a **horizontal plane** parallel to the \( xz \)-plane at \( y = 2 \).

### ii. Surface represented by \( z = 6 \)

Similarly, the equation \( z = 6 \) describes another **plane** in \( \mathbb{R}^3 \). Specifically:
- This plane is parallel to the \( xy \)-plane because it doesn't involve the \( x \) or \( y \) coordinates.
- The value of \( z \) is fixed at 6, so every point on this plane has the form \( (x, y, 6) \), where \( x \) and \( y \) can be any real numbers.

Thus, the surface is a **horizontal plane** parallel to the \( xy \)-plane at \( z = 6 \).

### Summary of Surfaces

1. The equation \( y = 2 \) represents a plane parallel to the \( xz \)-plane at \( y = 2 \).
2. The equation \( z = 6 \) represents a plane parallel to the \( xy \)-plane at \( z = 6 \).

Would you like further details or have any questions?

Here are some related questions:
1. What is the geometric interpretation of \( x = c \) in \( \mathbb{R}^3 \)?
2. How do you describe the intersection of two planes in \( \mathbb{R}^3 \)?
3. What kind of surface is represented by the equation \( x^2 + y^2 = 1 \) in \( \mathbb{R}^3 \)?
4. How can you find the line of intersection between the planes \( y = 2 \) and \( z = 6 \)?
5. What is the surface represented by \( x^2 + z^2 = 9 \) in \( \mathbb{R}^3 \)?

**Tip:** Visualizing equations in \( \mathbb{R}^3 \) often involves identifying which variables are fixed and which can vary, allowing you to describe the shape of the surface in space.

Explore the surfaces represented by the equations y = 2 and z = 6 in \( \mathbb{R}^3 \). Learn about their geometric interpretations, including their orientation and fixed coordinate values. Suitable for high school students studying coordinate geometry.

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