Volume of Liquid: Proving $\overrightarrow{v}\cdot\overrightarrow{n}$

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The discussion centers on demonstrating that the dot product $\overrightarrow{v} \cdot \overrightarrow{n}$ represents the volume of liquid flowing through a unit surface per unit time. The vector $\overrightarrow{v}$ denotes the uniform velocity of the liquid, while $\overrightarrow{n}$ is the unit normal vector to the surface. Participants confirm that referencing the "Volumetric flow rate" section on Wikipedia provides the necessary justification for this relationship, particularly the explanation regarding the dot product.

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mathmari
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Hey! :o

A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$.

Let $\overrightarrow{n}$ an unit vector perpendicular to the plane.

Show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volume of the liquid that passes through the unit surface of the plane in the unit of time.

Could you give me some hints how we could show this?? (Wondering)
 
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I found the following:

Volumetric flow rate - Wikipedia, the free encyclopedia

To show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volume of the liquid that passes through the unit surface of the plane in the unit of time, do we use the justification at the part "The reason for the dot product is as follows" of wikipedia?? (Wondering)
 
Yep. (Nod)
 

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