How is Torque Defined?

The torque \( \class{green}{\boldsymbol{M}} \) is defined as follows:

The torque is the cross product of lever-arm length \( \boldsymbol{r} \) and force \( \boldsymbol{F} \).

Because of the cross product the torque vector is always perpendicular to \( \boldsymbol{r} \) and \( \boldsymbol{F} \)! Its magnitude \( |\boldsymbol{r}\times \boldsymbol{F}| = \class{green}{M}\) is the area spanned by the vectors \( \boldsymbol{r} \) and \( \boldsymbol{F} \).

The magnitude of the torque, according to the definition of the cross product, is given by the following formula:

Here \(\varphi\) is the angle enclosed by the vectors \( \boldsymbol{r} \) and \(\boldsymbol{F}\). The torque becomes maximum when the two vectors are orthogonal to each other. So if the angle is \(\varphi = 90^{\circ}\). Formula 2 then simplifies to the product of the force and lever-arm length.