Question

in the cube shown below, which lines are skew?

cube - diagram with vertices labeled r, s, t, u, v, w, x, y

\\(\overrightarrow{tu}\\) and \\(\overrightarrow{sw}\\) \\(\overrightarrow{xy}\\) and \\(\overrightarrow{uy}\\) \\(\overrightarrow{rs}\\) and \\(\overrightarrow{sw}\\) \\(\overrightarrow{rs}\\) and \\(\overrightarrow{st}\\)

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Explanation:

Step1: Recall skew - lines definition

Skew lines are non - parallel and non - intersecting lines in 3 - D space.

Step2: Analyze each option

• For $\overrightarrow{TU}$ and $\overrightarrow{SW}$: $\overrightarrow{TU}$ and $\overrightarrow{SW}$ are in the same plane (the top - face of the cube), so they are not skew.
• For $\overrightarrow{XY}$ and $\overrightarrow{UY}$: $\overrightarrow{XY}$ and $\overrightarrow{UY}$ intersect at point $Y$, so they are not skew.
• For $\overrightarrow{RS}$ and $\overrightarrow{SW}$: $\overrightarrow{RS}$ and $\overrightarrow{SW}$ are in the same plane (the top - face of the cube), so they are not skew.
• For $\overrightarrow{RS}$ and $\overrightarrow{ST}$: $\overrightarrow{RS}$ and $\overrightarrow{ST}$ are in the same plane (the top - face of the cube), so they are not skew.
• Consider $\overrightarrow{RS}$ and $\overrightarrow{VX}$. $\overrightarrow{RS}$ is on the top - face and $\overrightarrow{VX}$ is on the bottom - face. They are non - parallel and non - intersecting. In general, if we consider the pairs given in the options, we find that $\overrightarrow{XY}$ and $\overrightarrow{UY}$ intersect, $\overrightarrow{TU}$ and $\overrightarrow{SW}$ are coplanar, $\overrightarrow{RS}$ and $\overrightarrow{SW}$ are coplanar, $\overrightarrow{RS}$ and $\overrightarrow{ST}$ are coplanar. But if we assume a correct non - given pair like a line on the top - face and a non - parallel non - intersecting line on the bottom - face, they would be skew. Since we have to choose from the given options, there is an error in the problem setup as none of the given pairs are skew. But if we go by the concept, skew lines are non - coplanar, non - parallel and non - intersecting.

Answer:

None of the given pairs are skew.