Let $\overrightarrow F $ be the force acting on a particle having position vector $\overrightarrow r ,$ and $\overrightarrow \tau $ be the torque of this force about the origin. Then
$\overrightarrow {r.} \overrightarrow \tau = 0\,\,$ and $\overrightarrow {F.} \overrightarrow \tau \ne 0\,\,$
$\overrightarrow {r.} \vec \tau \ne 0{\mkern 1mu} {\mkern 1mu} $ and $\overrightarrow {F.} \overrightarrow \tau = 0\,\,$
$\overrightarrow {r.} \vec \tau \ne 0{\mkern 1mu} $ and $\overrightarrow {F.} \overrightarrow \tau \ne 0$
$\overrightarrow {r.} \vec \tau = 0{\mkern 1mu} $ and $\overrightarrow {F.} \overrightarrow \tau = 0\,\,$
