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.