1. **Stating the problem:** We are asked to analyze pairs of vectors in a triangular prism: $\overrightarrow{DF}$ and $\overrightarrow{EH}$, $\overrightarrow{GI}$ and $\overrightarrow{DE}$, $\overrightarrow{DG}$ and $\overrightarrow{FI}$, $\overrightarrow{DF}$ and $\overrightarrow{DG}$. We want to understand their relationships, such as whether they are parallel, perpendicular, or neither. 2. **Recall vector properties:** - Two vectors are **parallel** if one is a scalar multiple of the other. - Two vectors are **perpendicular** if their dot product is zero. 3. **Step-by-step analysis:** - $\overrightarrow{DF}$ and $\overrightarrow{EH}$: These vectors connect vertices on opposite triangular bases. Since $D$ and $E$ are corresponding vertices on the two bases, and $F$ and $H$ are also corresponding vertices, $\overrightarrow{DF}$ and $\overrightarrow{EH}$ are parallel edges of the prism. - $\overrightarrow{GI}$ and $\overrightarrow{DE}$: $G$ and $D$ are vertices on the bottom base, $I$ is inside the prism connected to $G$ and $E$. $\overrightarrow{GI}$ and $\overrightarrow{DE}$ are not parallel and not perpendicular in general; their relationship depends on the exact coordinates. - $\overrightarrow{DG}$ and $\overrightarrow{FI}$: $D$ and $G$ are vertices on the bottom base, $F$ and $I$ are on the top base and inside the prism respectively. These vectors are not parallel or perpendicular in general. - $\overrightarrow{DF}$ and $\overrightarrow{DG}$: Both start at $D$; $F$ and $G$ are vertices on the bottom base. These vectors form edges of the base triangle and are not parallel; their dot product can be computed if coordinates are known. 4. **Summary:** Without explicit coordinates, we conclude: - $\overrightarrow{DF}$ and $\overrightarrow{EH}$ are parallel. - The other pairs are neither parallel nor perpendicular in general.