The probability that a molecule has its velocity component in the x-direction between $v_x$ and $v_{x+dx}$, in the y-direction between $v_y$ and $v_{y+dy}$, and in the z-direction between $v_z$ and $v_{z+dz}$ will be given by: \begin{equation} dN_{v_x,v_y,v_z}/N=g(v_x)g(v_y)g(v_z)dv_xdv_ydv_z \end{equation} $dN_{v_x,v_y,v_z}/N$ represents the probability that a molecule (fraction of molecules) has the end of its velocity vector inside a box with sides $dv_x, dv_y, dv_z$

Substituting the expressions computed above for the distribution functions in each spatial direction: \begin{equation} \phi(\bar{v})=g(v_x)g(v_y)g(v_z)=\left(\frac{m }{2\pi kT}\right)^{1/2}e^{-mv^2/2kT} \end{equation} Where, $v^2=v_x^2+v_y^2+v_z^2$ , the square of the magnitude of the velocity. This new distribution function does not depend on the orientation of the vector, but exclusively on its magnitude.