die Vene
vein
A blood vessel in the body that carries blood from the various parts of the body back to the heart.
Die Krankenschwester fand auf Anhieb eine Vene für die Blutabnahme.
The nurse found a vein for the blood draw right away.

vein
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vein
A blood vessel in the body that carries blood from the various parts of the body back to the heart.
Die Krankenschwester fand auf Anhieb eine Vene für die Blutabnahme.
The nurse found a vein for the blood draw right away.

'Vene' (vein) is a technical term from medicine and biology. In everyday life, you usually hear the word in the context of doctor visits, for example during a blood draw or infusion.
Für die Blutspende wird eine große Vene am Arm gesucht.
For the blood donation, they look for a large vein in the arm.
In medical contexts, there are fixed combinations. Common ones are 'Blut aus der Vene abnehmen' (to draw blood from the vein), 'eine Nadel in die Vene legen' (to insert a needle into the vein), 'eine Vene finden' (to find a vein), or 'eine Vene treffen' (to hit a vein).
Der Arzt musste die Nadel in eine Vene am Handrücken legen.
The doctor had to insert the needle into a vein on the back of the hand.
The related adjective is 'venös' (venous). It means 'relating to the veins' or 'through a vein'. Knowing this connection helps with understanding medical texts.
Der Arzt erklärte, dass es sich um eine venöse Blutung handelte.
The doctor explained that it was venous bleeding.
vessel / container
'das Gefäss' and 'die Vene' can both occur in the body and have to do with blood. 'Gefäss' is the general term for a tube-like structure through which fluid is transported. A blood vessel can be, for example, a vein, an artery, or a very small capillary. 'Vene' is more precise and means a blood vessel that carries blood back to the heart. The words can be confused because a vein is a type of vessel. If you know the exact type, you say 'Vene'. If you are speaking generally about such structures in the body, 'Gefäss' is more appropriate.

Bei der Blutabnahme sucht die Krankenschwester eine Vene am Arm.
During the blood draw, the nurse looks for a vein in the arm.
Das kleine Gefäss im Auge ist geplatzt, deshalb ist es rot.
The small vessel in the eye has burst, which is why it is red.