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I'm reading a section from mechanics about vectors and how to use cross products to describe area. One of the examples talked about calculating the rate in which water in a stream flows through the wire loop of a given area. It says if the plane is parallel to the flow, the flow is zero. Why is the flow zero?

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Are you visualising the loop immersed in the water? Suppose there were specks of some solid, carried along by the water. None of these specks would go through the loop, would it? The water would be flowing past the loop, not through it.

In general you calculate the volume flowing through the loop by taking the dot product of the area vector (a vector of magnitude equal to the loop area and direction along a normal to the loop) and the velocity vector of the water (assuming it's constant across the loop). If the plane of the loop is parallel to the flow then the normal to the loop is at right angles to the velocity vector. So the dot product is zero – in accordance with our intuitive reasoning above.

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