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 line equation.
The line is given in symmetric form: \[\n\frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0}\n\] Since the denominator for \( z \) is \( 0 \), \( z \) is constant. From: \[\n\frac{z-1}{0} = k\n\quad \Rightarrow \quad\nz-1 = 0k\n\quad \Rightarrow \quad\nz=1\n\] Therefore, \( z=1 \) for all points on the line.
Step 2: Analyze the direction.
Since \( z \) is constant, the line lies in the plane \( z=1 \). The direction ratios are: \[\n(3, 1, 0)\n\] This means the line moves in \( x \) and \( y \) but not 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.