The reason for the dot product is as follows. The only volume flowing through the cross-section is the amount normal to the area, that is, parallel to the unit normal. This amount is
where  is the angle between the unit normal  and the velocity vector  of the substance elements. The amount passing through the cross-section is reduced by the factor . As  increases less volume passes through. Substance which passes tangential to the area, that is perpendicular to the unit normal, does not pass through the area. This occurs when  and so this amount of the volumetric flow rate is zero:
These results are equivalent to the dot product between velocity and the normal direction to the area.