To find the parametric equation of a line passing through the point $(-1, 1, -1)$ and parallel to the vector $(-1, 1, -1)$, let's go through the following steps:

1. The general form of the parametric equation for a line in three-dimensional space is: $(x, y, z) = (x_0, y_0, z_0) + t(a, b, c)$ where $(x_0, y_0, z_0)$ is a point on the line, and $(a, b, c)$ is a direction vector parallel to the line.

2. In this problem:

   • The point $(x_0, y_0, z_0) = (-1, 1, -1)$.
   • The direction vector $(a, b, c) = (-1, 1, -1)$.

3. Using this information, we can write the parametric equations as: $x = -1 - t$ $y = 1 + t$ $z = -1 - t$

4. Comparing these equations to the given options:

   • The correct answer is option (d).

Would you like further details on parametric equations or related topics?

Here are five related questions to expand on this:

1. How do you find the direction vector for a line given two points on it?
2. What is the significance of the parameter $t$ in parametric equations?
3. How can you verify if two lines in 3D space are parallel?
4. How would the parametric equation change if the direction vector was different?
5. How can parametric equations be converted to symmetric form?

Tip: Always check both the point and direction vector when writing parametric equations to ensure accuracy.