Determine the equation of the line parallel to the y-axis through the point p (3,2,0) 

To find the equation of a line that is parallel to the y-axis and passes through the point \( P(3, 2, 0) \) in three-dimensional space, you need to fix the \( x \)- and \( z \)-coordinates to those of the given point and let the \( y \)-coordinate vary.

Since the line is parallel to the y-axis:
- \( x \) will remain constant at 3.
- \( z \) will remain constant at 0.
- \( y \) will vary.

The parametric equations of the line can be written as:
\[
x = 3, \quad y = t, \quad z = 0
\]
where \( t \) is a parameter that can take any real value.

Thus, the equation of the line is:

\[
\boxed{x = 3, \quad z = 0}
\]

This describes a line parallel to the y-axis passing through the point \( (3, 2, 0) \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the equation of a line parallel to the z-axis through the point \( (3, 2, 0) \)?
2. How would you describe a line parallel to the x-axis passing through the point \( (3, 2, 0) \)?
3. How can you determine if two lines in 3D space are parallel?
4. What is the equation of a line parallel to the y-axis through a different point, say \( (1, 4, -2) \)?
5. How do you find the intersection of two lines in 3D space?

**Tip:** In 3D geometry, when given a line parallel to one of the axes, only the coordinate corresponding to that axis is allowed to vary; the others remain fixed at the point's coordinates.

Learn how to find the equation of a line parallel to the y-axis passing through the point (3, 2, 0) in 3D space. Understand the concept of parametric equations and fixed coordinates in 3D geometry. Suitable for high school students studying 3D geometry.

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