To determine which combinations of axial and equatorial bonds in dimethyl cyclohexane show cis or trans orientations, we will analyze the structure of dimethyl cyclohexane and the implications of the axial and equatorial positions of the methyl groups.
### Step-by-Step Solution:
1. **Understanding Dimethyl Cyclohexane Structure**:
- Dimethyl cyclohexane consists of a cyclohexane ring with two methyl (–CH₃) groups attached to it. The cyclohexane can adopt a chair conformation, which is the most stable form.
2. **Chair Conformation**:
- In the chair conformation, substituents can occupy two types of positions: axial (A) and equatorial (E). Axial positions are perpendicular to the plane of the ring, while equatorial positions are roughly parallel to the plane of the ring.
3. **Identifying Cis and Trans Configurations**:
- **Cis Configuration**: Both methyl groups are on the same side of the cyclohexane ring (both equatorial or both axial).
- **Trans Configuration**: The methyl groups are on opposite sides of the cyclohexane ring (one axial and one equatorial).
4. **Evaluating Each Combination**:
- **(i) 1E, 2E**: Both methyl groups are equatorial. This is a **cis** configuration.
- **(ii) 1E, 3E**: One methyl group is equatorial at carbon 1 and the other at carbon 3. This is a **trans** configuration.
- **(iii) 1E, 4E**: Both methyl groups are equatorial at carbons 1 and 4. This is a **cis** configuration.
- **(iv) 1E, 2A**: One methyl group is equatorial and the other is axial. This is a **trans** configuration.
- **(v) 1E, 3A**: One methyl group is equatorial and the other is axial. This is a **trans** configuration.
- **(vi) 1E, 4A**: One methyl group is equatorial and the other is axial. This is a **trans** configuration.
- **(vii) 1A, 3A**: Both methyl groups are axial. This is a **cis** configuration.
5. **Conclusion**:
- The combinations that show **cis** orientation are:
- (i) 1E, 2E
- (iii) 1E, 4E
- (vii) 1A, 3A
- The combinations that show **trans** orientation are:
- (ii) 1E, 3E
- (iv) 1E, 2A
- (v) 1E, 3A
- (vi) 1E, 4A
### Final Answer:
The combinations that show cis orientation are (i) 1E, 2E and (iii) 1E, 4E, while the combinations that show trans orientation are (ii) 1E, 3E, (iv) 1E, 2A, (v) 1E, 3A, and (vi) 1E, 4A.
