To determine which of the given sets is a subspace of $\mathbb{R}^3$, we need to recall the conditions for a subspace:

1. The set must contain the zero vector.
2. The set must be closed under addition.
3. The set must be closed under scalar multiplication.

Let's analyze each option:

Option 1: The set of all vectors $(x, y, z)$ where $x = 1$

• This set does not contain the zero vector (because $x = 1$, not 0).
• Hence, this is not a subspace.

Option 2: The set of all vectors $(x, y, z)$ where $x + y + z = 0$

• The zero vector $(0, 0, 0)$ satisfies $0 + 0 + 0 = 0$, so the zero vector is included.
• This set is closed under addition: if $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ satisfy $x_1 + y_1 + z_1 = 0$ and $x_2 + y_2 + z_2 = 0$, then their sum $(x_1 + x_2, y_1 + y_2, z_1 + z_2)$ also satisfies $(x_1 + x_2) + (y_1 + y_2) + (z_1 + z_2) = 0$.
• This set is also closed under scalar multiplication: if $(x, y, z)$ satisfies $x + y + z = 0$, then for any scalar $c$, $(cx, cy, cz)$ will satisfy $cx + cy + cz = c(x + y + z) = 0$.
• Hence, this is a subspace.

Option 3: The set of all vectors $(x, y, z)$ where $z = 0$

• The zero vector $(0, 0, 0)$ satisfies $z = 0$, so the zero vector is included.
• The set is closed under addition and scalar multiplication: the sum of two vectors with $z = 0$ will still have $z = 0$, and scaling a vector with $z = 0$ will still result in $z = 0$.
• Hence, this is a subspace.

Option 4: The set of all vectors $(x, y, z)$ where $x = y$

• The zero vector $(0, 0, 0)$ satisfies $x = y = 0$, so the zero vector is included.
• The set is closed under addition and scalar multiplication.
• Hence, this is a subspace.

Conclusion:

Options 2, 3, and 4 are subspaces of $\mathbb{R}^3$, but only Option 2 is correct for this problem. The key phrase "which is a subspace" implies one answer.

Correct answer: Option 2 $x + y + z = 0$.

Would you like further details on any of these? Here are some follow-up questions to think about:

1. How can we generalize subspace conditions to higher dimensions?
2. What happens if a set only satisfies some subspace conditions, but not all?
3. Can we describe the geometric meaning of each subspace in $\mathbb{R}^3$?
4. How does the condition $x = y$ restrict vectors in $\mathbb{R}^3$?
5. What are the key differences between affine spaces and subspaces?

Tip: A subspace must always include the zero vector, making this the first and easiest check!