rüber
over
Colloquial short form of 'herüber' or 'hinüber', describing movement across a short distance or an imagined line.
Gib mir mal bitte das Salz rüber.
Please pass me the salt over here.

over
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over
Colloquial short form of 'herüber' or 'hinüber', describing movement across a short distance or an imagined line.
Gib mir mal bitte das Salz rüber.
Please pass me the salt over here.

'Rüber' (over) is the colloquial short form of 'herüber' (over here/across to here) or 'hinüber' (over there/across to there). It describes movement from one place to another, usually over a short distance.
Kommst du heute Abend zu mir rüber?
Are you coming over to my place tonight?
'Rüber' (over) often functions as a separable prefix for verbs like 'rüberkommen' (to come over), 'rübergehen' (to go over), or 'rüberreichen' (to pass over). In a main clause, the prefix stands at the end.
Kannst du mir bitte die Butter rüberreichen?
Can you please pass me the butter?
The separable verb 'etwas rüberbringen' (to get something across) has the figurative meaning of successfully communicating a message or feeling. What is communicated is the accusative object.
In seiner Rede hat er die Dringlichkeit des Problems gut rübergebracht.
In his speech, he got the urgency of the problem across well.
The expression 'rübermachen' (to go over) has a specific historical meaning. During the Cold War, it was the colloquial term for fleeing from East Germany (GDR) to West Germany.
Seine Familie hat kurz vor dem Mauerbau noch rübergemacht.
His family still managed to go over shortly before the Berlin Wall was built.
to mediate / arrange / convey
'vermitteln' and 'rüber' are connected here through the phrase 'etwas rüberbringen' (to get something across). Both can mean showing information, an idea, or a feeling in a way that others understand it. 'vermitteln' is more neutral and fits well with teaching, explanations, knowledge, or an impression. 'etwas rüberbringen' is more colloquial and emphasizes whether the message really reaches the others. In formal texts, 'vermitteln' is usually more appropriate; in conversation, 'rüberbringen' sounds more natural.

Bei der Präsentation kam seine Begeisterung gut rüber.
His enthusiasm came across well during the presentation.
Die Lehrerin vermittelt den Schülern die Regeln mit einfachen Beispielen.
The teacher teaches the students the rules using simple examples.