Line of action

For the simple geometry associated with the figure, there are three equivalent equations for the magnitude of the torque associated with a force ${\displaystyle {\vec {F}}}$  directed at displacement ${\displaystyle {\vec {r}}}$  from the axis whenever the force is perpendicular to the axis:

${\displaystyle {\begin{aligned}||{\vec {\tau }}||&=||{\vec {r}}\times {\vec {F}}||\\&=rF_{\perp }\\&=r_{\perp }F\\&=||rF\sin \theta ||\,,\end{aligned}}}$ 

where ${\displaystyle {\vec {r}}\times {\vec {F}}}$  is the cross-product, ${\displaystyle F_{\perp }}$  is the component of ${\displaystyle {\vec {F}}}$  perpendicular to ${\displaystyle {\hat {r}}}$ , ${\displaystyle r_{\perp }}$  is the moment arm, and ${\displaystyle \theta }$  is the angle between ${\displaystyle {\vec {r}}}$  and ${\displaystyle {\vec {F}}}$