My purple and red Dyson has become hard to push. Could a belt have broken?

It's simple to tell if a belt has broken. Just switch the machine on and have a look (look, don't feel) to see if the brush roller is rotating. If it is, then the belt is not broken.
Depending on how much use your machine has had, it's quite likely that the brushes on the brush roller are worn to the point where you need a new brush roller.  Most vacuum cleaners rely on rotation of the brush roller to pull them along, so if the bristles are too short this doesn't happen.
I hope this helps. Please remember to rate this response.

Try looking at the small rollers on the plate that surrounds the brushes at base of the machine they may be jammed and or the plate may need cleaning

The roller brush maybe worn down to the point that the brushes are not in contact with the floor. That will make it hard to push. Time to replace it. Or a belt maybe broken and not turning the roller. The belt will need to be replaced.