The straight line `(x-3)/3=(y-2)/1=(z-1)/0` is (A) Parallel to x-axis (B) Parallel to the y-axis (C) Parallel to the z-axis (D) Perpendicular to the z-axis

Options

A. (A) Parallel to x-axis
B. (B) Parallel to the y-axis
C. (C) Parallel to the z-axis
D. (D) Perpendicular to the z-axis

Explanation

To determine the nature of the straight line given by the equation \((x-3)/3 = (y-2)/1 = (z-1)/0\), we can analyze the direction ratios of the line.

### Step-by-Step Solution:

1. **Identify the Direction Ratios**:
The line is given in the symmetric form:
\[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \]
From this, we can extract the direction ratios:
- For \(x\): The direction ratio is \(3\).
- For \(y\): The direction ratio is \(1\).
- For \(z\): The direction ratio is \(0\).

Thus, the direction ratios of the line are \( (3, 1, 0) \).

2. **Analyze the Direction Ratios**:
- The direction ratio for \(z\) is \(0\), which indicates that the line does not change in the \(z\)-direction. Therefore, the line is not moving up or down in the \(z\)-axis.
- The non-zero direction ratios for \(x\) and \(y\) indicate that the line has components in the \(x\) and \(y\) directions.

3. **Determine the Orientation**:
- A line is said to be **parallel to the x-axis** if its direction ratios are of the form \((a, 0, 0)\).
- A line is said to be **parallel to the y-axis** if its direction ratios are of the form \((0, b, 0)\).
- A line is said to be **parallel to the z-axis** if its direction ratios are of the form \((0, 0, c)\).
- A line is **perpendicular to the z-axis** if the dot product of its direction vector and the direction vector of the z-axis (which is \(k\) or \((0, 0, 1)\)) is zero.

4. **Check Each Option**:
- **Option (A)**: Parallel to the x-axis. This is incorrect because the direction ratio for \(y\) is not zero.
- **Option (B)**: Parallel to the y-axis. This is incorrect because the direction ratio for \(x\) is not zero.
- **Option (C)**: Parallel to the z-axis. This is incorrect because the direction ratios for both \(x\) and \(y\) are not zero.
- **Option (D)**: Perpendicular to the z-axis. To check this, we calculate the dot product of the direction vector \((3, 1, 0)\) with the direction vector of the z-axis \((0, 0, 1)\):
\[ (3, 1, 0) \cdot (0, 0, 1) = 3 \cdot 0 + 1 \cdot 0 + 0 \cdot 1 = 0 \]
Since the dot product is zero, the line is indeed perpendicular to the z-axis.

### Conclusion:
The correct answer is **(D) Perpendicular to the z-axis**.