A line that is parallel to the y-axis in three-dimensional space (3D) means that the line runs parallel to the y-axis and does not change its $y$-coordinate. The general form of such a line can be expressed by fixing the $x$- and $z$-coordinates, while allowing the $y$-coordinate to vary.

Given the point $P\left(\frac{3}{2}, 0\right)$, the equation of the line parallel to the y-axis is:

$x = \frac{3}{2}, \quad z = 0$

The $y$-coordinate can take any value, so the parametric equations of the line are:

$x = \frac{3}{2}, \quad y = t, \quad z = 0$

where $t$ is a parameter that can take any real value.

Thus, the equation of the line is:

$\boxed{x = \frac{3}{2}, \quad z = 0}$

This describes a line parallel to the y-axis, passing through the point $\left(\frac{3}{2}, 0, 0\right)$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the equation of a line parallel to the x-axis in 3D space?
2. What is the equation of a line parallel to the z-axis passing through a different point, say $(1, 2, 3)$?
3. How does the concept of parallel lines differ in 2D and 3D space?
4. How would you find the distance between a point and a line in 3D space?
5. What is the general form of a line equation in 3D using direction ratios?

Tip: When working with lines in 3D, remember that fixing two coordinates allows the third to vary, describing lines parallel to one of the axes.