In this diagram of a rectangular prism, the task is to identify pairs of skew lines. Skew lines are lines that are neither parallel nor do they intersect. In 3D space, skew lines exist on different planes.

Let’s break down each pair provided as options:

1. $\overrightarrow{PT}$ and $\overrightarrow{RU}$:

   • $\overrightarrow{PT}$ runs vertically from point P to point T.
   • $\overrightarrow{RU}$ runs diagonally across the top face from R to U.
   • These lines do not lie on the same plane, and they neither intersect nor are parallel, so they are skew.

2. $\overrightarrow{RU}$ and $\overrightarrow{OP}$:

   • $\overrightarrow{RU}$ is the diagonal across the top face (R to U).
   • $\overrightarrow{OP}$ is the diagonal on the front face (O to P).
   • Since these lines are not on the same plane, they are also skew.

3. $\overrightarrow{NQ}$ and $\overrightarrow{OP}$:

   • $\overrightarrow{NQ}$ is the edge on the left face from N to Q.
   • $\overrightarrow{OP}$ is the diagonal on the front face (O to P).
   • Again, these lines exist on different planes and do not intersect or run parallel, making them skew.

4. $\overrightarrow{RS}$ and $\overrightarrow{NO}$:

   • $\overrightarrow{RS}$ is the horizontal edge on the right side from R to S.
   • $\overrightarrow{NO}$ is the bottom edge from N to O.
   • These lines lie on different planes and are skew.

The correct answer from the options provided is $\overrightarrow{PT}$ and $\overrightarrow{RU}$ because they are clearly skew, residing on different planes and not intersecting or parallel.

Let me know if you need further details or clarifications!

Follow-up Questions:

1. Can you explain why two lines on the same face of a prism are not skew?
2. What are some key differences between skew lines and parallel lines in 3D geometry?
3. Can skew lines exist in two-dimensional space? Why or why not?
4. How would the concept of skew lines change if we were dealing with a different polyhedron, like a pyramid?
5. What are some real-world examples of skew lines?

Tip: In 3D geometry, it’s helpful to visualize lines on different planes to understand their relationships better.