# How is Torque Defined?

Level 3 (with higher mathematics)

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

Definition of the torque
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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:

Magnitude of the torque
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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.

Maximum torque
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