The line x = 1, y = 2 is

Options

A. parallel to the x-axis
B. parallel to the y-axis
C. parallel to the z-axis
D. parallel to the xy-plane

Answer

To determine the nature of the line given by the equations \( x = 1 \) and \( y = 2 \), we can follow these steps:

### Step 1: Understand the given equations
The equations \( x = 1 \) and \( y = 2 \) indicate that the line is defined at a specific point in the 3D space where \( x \) and \( y \) are constant.

**Hint:** Recognize that these equations represent fixed values for \( x \) and \( y \) while \( z \) can take any value.

### Step 2: Rewrite the equations in a standard form
We can express the equations in a parametric form. Since \( x \) and \( y \) are constant, we can let \( z \) be a parameter \( t \). Thus, we can write:
- \( x = 1 \)
- \( y = 2 \)
- \( z = t \)

**Hint:** Identify the parameter that allows the third dimension to vary freely.

### Step 3: Identify the direction vector
From the parametric equations, we can see that the direction vector of the line can be represented as:
\[ \vec{d} = (0, 0, 1) \]
This indicates that the line moves in the \( z \)-direction only.

**Hint:** The direction vector shows how the line extends in space. If only one component is non-zero, it indicates movement along that axis.

### Step 4: Determine the nature of the line
Since the line is defined by constant \( x \) and \( y \) values, and \( z \) can vary, this means the line is vertical and parallel to the \( z \)-axis.

**Hint:** Consider how the line behaves in relation to the coordinate planes.

### Step 5: Conclusion
The line described by \( x = 1 \) and \( y = 2 \) is parallel to the \( z \)-axis and lies in the plane where \( x = 1 \) and \( y = 2 \). Therefore, it does not lie in the \( xy \)-plane but extends vertically.

**Final Answer:** The line is parallel to the \( z \)-axis.

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