Is (1R,2S)‐1,2‐dimethylcyclohexane superimposable on its mirror image? What about (1R,2R)‐1,2‐dimethylcyclohexane?
In general, any two stereoisomers that are not mirror images are diastereomers, More specifically, these are geometrical isomers or cis–trans isomers. (1R,2S)-1,2-Dimethylcyclohexane is the cis isomer, whereas (1R,2R)-1,2-dimethylcyclohexane is the trans isomer.
For the second part of your question, (1R,2S)-1,2-dimethylcyclohexane is superimposable on its mirror image as it has a plane of symmetry. (1R,2R)-1,2-Dimethylcyclohexane is not superimposable as there are no elements of symmetry and hence is optically active.
The relation between them is Optical isomers. The relation between them is Geometrical isomers also. Any 2 compound showing either of GI or OI are related as diastreomer.
In more appropriate words diastreoismer is used to relate the streoisomers which are non superimpose mirror images of each other. Hence they are also related as diasterisomers
Is (1R,2S)‐1,2‐dimethylcyclohexane superimposable on its mirror image? It is superimposable on its mirror image.The reason being is the presence of events of symmetry POS. You can verify this even by drawing the mirror image of the compound and try to superimpose on the given structure .They will perfectly overlap (and they would be related as *identical *)
,> What about (1R,2R)‐1,2‐dimethylcyclohexane? It is clearly non superimposable on its mirror image as there is absence of any of the elements of symmetry like POS and COS . You can also verify this by drawing the mirror image of the compound and try to superimpose on the given structure. They would not overlap.. (its mirror image would be related as Enantiomer with it.)