I need to express force exerted on a body due to aerodynamic drag mathematically as vector in matrix form. Let $F\_f$ be the exerted force, $k\_{lf}$ the aerodynamic constant and $\vec{V} = [u, v, w]^T$ the velocity vector. $[u, v, w]^T$ can either be positive or negative and $F\_f$ is proportional to $\vec{V}^2$ but directed opposite. Writing:

$F\_f = -k\_{lf} \begin{bmatrix} u^2 \\ v^2 \\ w^2 \end{bmatrix}$

is wrong, as $u^2$ is always positive. What's the correct form of the above equation in matrix form, to preserve $u, v, w$ sign? Is using $sgn(x)$ the right choice?

PS. Does wrinting $\vec{V}^2$ preserve sign, i.e. is the below true?

$\vec{V}^2 = \begin{bmatrix} sgn(u) u^2 \\ sgn(v) v^2 \\ sgn(w) w^2\end{bmatrix}$