Torque is mathematically defined as the cross product of the position vector ($\vec{r}$) and the force vector ($\vec{F}$), i.e., $\vec{\tau} = \vec{r} \times \vec{F}$. A fundamental property of the cross product is that the resulting vector ($\vec{\tau}$) is always perpendicular (orthogonal) to the plane containing both $\vec{r}$ and $\vec{F}$. Consequently, $\vec{\tau}$ is perpendicular to $\vec{r}$ and also perpendicular to $\vec{F}$. Since the dot product of any two perpendicular vectors is zero, we must have $\vec{r} \cdot \vec{\tau} = 0$ and $\vec{F} \cdot \vec{\tau} = 0$.