1. to have a vision, to picture ("jemandem (Dative) schwebt etwas vor", - note that German has the roles backwards. So the idea is the subject and the person doing the imagining is the Dative object. Think of it as something hovering in front of your eyes.)
Meanings
Examples
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A quick look at the meaning of "schweben", where it comes from and what it has to do with imagination and blabbering.
Vocab:
schweben, vorschweben, der Schweif, abschweifen, ohne Umschweife

Word Family
Root: *swei-