Question

The equation of a line passing through the points (1,0,-2) and parallel to z-axis is:

Answer 1

$\frac{x - 1}{0} = \frac{y}{0} = \frac{z + 2}{1}$

Answer 2

To find the equation of a line passing through a given point and parallel to an axis, we use the symmetric form of the line equation in 3D space.

1. General Equation of a Line:
The equation of a line passing through a point $(x_1, y_1, z_1)$ and having direction ratios $(a, b, c)$ is given by:

$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$

2. Identify the Given Point:
The line passes through the point (1, 0, -2).
So, $(x_1, y_1, z_1) = (1, 0, -2)$.

3. Determine the Direction Ratios:
The line is parallel to the z-axis. The direction vector of the z-axis is $\vec{k} = (0, 0, 1)$.
Therefore, the direction ratios of the required line are $(a, b, c) = (0, 0, 1)$.

4. Substitute the Values:
Substitute the point $(1, 0, -2)$ and the direction ratios $(0, 0, 1)$ into the general equation:

$$\frac{x - 1}{0} = \frac{y - 0}{0} = \frac{z - (-2)}{1}$$ $$\frac{x - 1}{0} = \frac{y}{0} = \frac{z + 2}{1}$$

This equation implies that $x - 1 = 0$ (i.e., $x = 1$) and $y = 0$, while $z$ can take any real value. This defines a line parallel to the z-axis, passing through the point (1, 0, -2).

The equation of the line is $\frac{x - 1}{0} = \frac{y}{0} = \frac{z + 2}{1}$.