The equations of x- axis in space are

Options

A. x=0, y=0
B. x=0 , z=0
C. x=0
D. y=0 , z=0

Answer

To find the equations of the x-axis in space \( \mathbb{R}^3 \), we need to understand the characteristics of points that lie on the x-axis.

### Step-by-step solution:

1. **Understanding the x-axis**:
- The x-axis in three-dimensional space is defined as the set of all points where the y-coordinate and z-coordinate are both zero. This means that any point on the x-axis can be represented as \( (x, 0, 0) \), where \( x \) can take any real value.

2. **Setting up the equations**:
- From the definition above, we can derive the equations that represent the x-axis:
- \( y = 0 \) (since all points on the x-axis have a y-coordinate of 0)
- \( z = 0 \) (since all points on the x-axis have a z-coordinate of 0)

3. **Combining the equations**:
- Therefore, the equations of the x-axis can be expressed as:
\[ y = 0 \quad \text{and} \quad z = 0 \]

4. **Identifying the correct option**:
- Among the given options, we need to find which one correctly represents these conditions:
- Option A: \( x = 0, y = 0 \) (incorrect, as x can be any value)
- Option B: \( x = 0, z = 0 \) (incorrect, as x can be any value)
- Option C: \( x = 0 \) (incorrect, as y and z must also be 0)
- Option D: \( y = 0 \) and \( z = 0 \) (correct, as this matches our findings)

Thus, the correct answer is **Option D: \( y = 0 \) and \( z = 0 \)**.