headwind
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against the wind
Describes a movement or state that occurs directly against the wind direction, or figuratively against resistance. This adjectival or adverbial use is very rare and unusual; instead, the noun form 'der Gegenwind' (the headwind) or the phrase 'gegen den Wind' (against the wind) is used.
Obwohl sie die Strecke fast nur gegenwind fuhren, erreichten sie eine gute Zeit.
Although they rode almost the entire route against the wind, they achieved a good time.
The word is simply made up of 'gegen' (against) and 'Wind' (wind). So the meaning is literal: in the direction from which the wind is coming.
Die Radtour war anstrengend, weil wir die ganze Zeit Gegenwind hatten.
The bike ride was exhausting because we had a headwind the whole time.
Use as a single adjective or adverb is extremely rare. Instead, people almost always use the phrase 'gegen den Wind' (against the wind) or the noun 'der Gegenwind' (the headwind).
Wir sind gegen den Wind gelaufen.
We walked against the wind.
The noun 'der Gegenwind' (the headwind) is very often used figuratively for resistance, criticism, or difficulties. For example, you can say 'politischen Gegenwind bekommen' (to face political headwind/opposition).
Das neue Projekt bekam von Anfang an starken Gegenwind aus der Bevölkerung.
The new project faced strong opposition from the public from the very beginning.
opposite
'Gegenwind' and 'entgegengesetzt' can both express that something goes in the other direction. 'Gegenwind' is normally used as a noun: the wind comes from the front and makes movement harder. The adjectival or adverbial use is rare; it is usually more natural to say 'gegen den Wind' (against the wind). 'entgegengesetzt' is a normal adjective and generally describes two directions that run exactly against each other. For wind and outdoor movement, 'Gegenwind' (headwind) or 'gegen den Wind' fits. For directions, arrows, paths, or movements in general, 'entgegengesetzt' fits better.
Wegen des starken Gegenwinds kamen wir mit dem Fahrrad nur langsam voran.
Because of the strong headwind, we made only slow progress by bike.
Die beiden Autos fuhren in entgegengesetzte Richtungen.
The two cars drove in opposite directions.