Question

The straight line \[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \] is:

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

Hint:

When a coordinate denominator is \( 0 \) in a symmetric line equation, it means the corresponding coordinate is constant. The line is then parallel to the plane formed by the other two axes and perpendicular to the axis of the constant coordinate.

Answer 1

The Correct Option is D

Step 1: Understand the given line equation.
The line is given in symmetric form: \[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \] Since the denominator for \( z \) is \( 0 \), this indicates that \( z \) is constant. From: \[ \frac{z-1}{0} = k \quad \Rightarrow \quad z-1 = 0k \quad \Rightarrow \quad z=1 \] Thus, \( z \) is fixed at \( z=1 \) for all points on the line.
Step 2: Analyze the direction.
Since \( z \) is constant and does not vary, the line lies completely in the plane \( z=1 \). The direction ratios are: \[ (3, 1, 0) \] which means the line moves along \( x \) and \( y \) but has no movement in \( z \). Thus, the line is parallel to the x-y plane and perpendicular to the z-axis.
Step 3: Conclude the answer.
Hence, the line is perpendicular to the z-axis.