If $\vec{F}$ is the force acting on a particle having position vector $\vec{r}$ and $\vec{\tau }$ be the torque of this force about the origin, then:

Torque is defined as the cross product of the position vector and the force vector:
$$\vec{\tau }=\vec{r}\times \vec{F}$$
By the properties of the cross product:The resulting vector $\vec{\tau }$ is always perpendicular to both $\vec{r}$ and $\vec{F}$.If two vectors are perpendicular, their dot product is zero.Therefore:
$$\vec{r}\cdot \vec{\tau }=0$$
$$\vec{F}\cdot \vec{\tau }=0$$