The question is asking for the parametric equation of a line that passes through the points $(1, 2, 3)$ and $(3, 2, 4)$.

To find the parametric equations of the line, follow these steps:

Step 1: Find the direction vector

The direction vector $\mathbf{v}$ can be found by subtracting the coordinates of the two points: $\mathbf{v} = (3 - 1, 2 - 2, 4 - 3) = (2, 0, 1)$

Step 2: Write the parametric equations

The parametric equations of the line through point $(x_0, y_0, z_0)$ with direction vector $(v_x, v_y, v_z)$ are: $x = x_0 + v_x t$ $y = y_0 + v_y t$ $z = z_0 + v_z t$ Here, the point $(1, 2, 3)$ will be used as $(x_0, y_0, z_0)$, and the direction vector $(2, 0, 1)$ will be used.

So the parametric equations are: $x = 1 + 2t, \quad y = 2 + 0t = 2, \quad z = 3 + t$

Thus, the correct answer is: $x = 1 + 2t, \quad y = 2, \quad z = 3 + t$

Answer:

The correct option is the one where: $x = 1 + 2t, \quad y = 2, \quad z = 3 - t$

Would you like more details about this problem or any clarifications?

Here are 5 follow-up questions for you:

1. What is a direction vector, and how do we find it?
2. How can we determine if two points are collinear?
3. What happens if the direction vector is zero?
4. Can parametric equations be used to represent curves, and how?
5. How do we find the parametric equation of a line in three-dimensional space?

Tip: Double-check the signs and terms in parametric equations—small errors can lead to incorrect answers.