Pressure is a compressive force per unit area, and it gives the impression of being a vector. However, pressure at any point in a fluid is the same in all directions. That is, it has magnitude but not a specific direction, and thus it is a scalar quantity. This can be demonstrated by considering a small wedge-shaped fluid element of unit length (into the page) in equilibrium, as shown:
The mean pressures at the three surfaces are P1, P2, and P3, and the force acting on a surface is the product of mean pressure and the surface area. From Newton's second law, a force balance in the x- and z-directions gives:
where ρ is the density and W = mg = ρg Δx Δz/2 is the weight of the fluid element. Noting that the wedge is a right triangle, Δx = L cos θ and Δz = L sin θ. Substituting these geometric relations and dividing Eq1 by Δz and Eq2 by Δx gives:
The last term in Eq4 drops out as Δz → 0 and the wedge becomes infinitesimal, and thus the fluid element shrinks to a point. Then combining the results of these two relations gives:
P1 = P2 = P3 = P
regardless of the angle θ. This analysis can be repeated for an element in the xz–plane and a similar result will be obtained. Thus it is concluded that the pressure at a point in a fluid has the same magnitude in all directions. It can be shown in the absence of shear forces that this result is applicable to fluids in motion as well as fluids at rest.
