Consider a screw $\{ {\cal V} \}$ (of resultant ${\bf V}$) in referential $\cal E$ which satisfies the following property:
\[
{\bf \hat{e}}_3 \cdot {\bf v}_P = 0
\]
where ${\bf \hat{e}}_3$ is a unit vector of $\cal E$. Then $\{ {\cal V} \}$ is either a couple or a slider.
For any two points $P$ and $Q$, we have ${\bf \hat{e}}_3 \cdot ({\bf v}_Q - {\bf v}_P)= 0$. Then we can write
${\bf \hat{e}}_3 \cdot ( {\bf V} \times {\bf r}_{PQ} ) =0$. This gives the triple scalar product
$({\bf \hat{e}}_3 , {\bf V} , {\bf r}_{PQ} ) =0$. Since this is true for all $P$ and $Q$, this implies that ${\bf \hat{e}}_3 \times {\bf V} = {\bf 0}$. We then deduce that the screw satisfies
\[
{\bf V} \cdot {\bf v}_P = 0
\]
A screw whose scalar invariant ${\bf V} \cdot {\bf v}_P$ vanishes is either a couple or a slider.
From this result, we conclude that the kinematic screw of a body in planar motion is either a couple or a slider at any given instant: the body is in instantaneous translation or in instantaneous rotation.
