saugen
to vacuum
To remove dust and dirt from a surface with a vacuum cleaner.
Ich muss am Wochenende unbedingt die Wohnung saugen.
I really have to vacuum the apartment this weekend.

to vacuum / to suck
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to vacuum
To remove dust and dirt from a surface with a vacuum cleaner.
Ich muss am Wochenende unbedingt die Wohnung saugen.
I really have to vacuum the apartment this weekend.

The verb 'saugen' (to vacuum / to suck) has two past participle forms, and their use depends on the meaning. For vacuuming, the weak form 'gesaugt' is used. In all other meanings, the strong form 'gesogen' is used.
Ich habe gestern die ganze Wohnung gesaugt.
I vacuumed the whole apartment yesterday.
When 'saugen' (to vacuum) is used without an object or only with a location, it almost always means 'Staub saugen' (to vacuum). The word 'Staub' (dust) is often left out in everyday language.
Kannst du bitte im Wohnzimmer saugen?
Could you please vacuum in the living room?
When a person or animal sucks on something, the preposition 'an' is used with the dative. Typical phrases are 'an einem Strohhalm saugen' (to suck on a straw) or 'am Daumen saugen' (to suck one's thumb).
Der Kolibri saugt Nektar an der Blüte.
The hummingbird sucks nectar from the flower.
The idiom 'sich etwas aus den Fingern saugen' (to make something up out of thin air) is fixed. The person who invents something is in the dative ('sich'), and the invented information is in the accusative.
Diese Ausrede hast du dir doch komplett aus den Fingern gesogen.
You completely made up that excuse out of thin air.
In the meaning 'sich festsaugen' (to suction oneself firmly), the verb is used reflexively with the preposition 'an' and a dative or accusative object to name the surface.
Der Oktopus kann sich mit seinen Armen an Felsen saugen.
The octopus can suction itself to rocks with its arms.
to pull / move
'ziehen' and 'saugen' both have to do with movement toward oneself. 'ziehen' is general: you move something toward you or along behind you by using force. 'saugen' is more specific: air or liquid is drawn inward through a narrow opening, often with the mouth. That is why you can pull on a door, but suck juice through a straw. The words can be confused because with 'saugen', a kind of pulling also happens. Even so, 'saugen' only sounds natural when suction, the mouth, or a narrow opening is important.

Das Kind saugt den Saft langsam durch den Strohhalm.
The child slowly sucks the juice through the straw.
Bitte zieh den schweren Koffer nicht über den frischen Boden.
Please don't pull the heavy suitcase across the new floor.