If $\overrightarrow{\mathrm{F}}$ is the force acting on a particle having position vector $\vec{r}$ and…
If $\overrightarrow{\mathrm{F}}$ 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
$\overrightarrow{\mathrm{r}} \cdot \vec{\tau} \neq 0$ and $\overrightarrow{\mathrm{F}} \cdot \vec{\tau}=0$
$\overrightarrow{\mathrm{r}} \cdot \vec{\tau}=0$ and $\overrightarrow{\mathrm{F}} \cdot \vec{\tau}=0$
$\overrightarrow{\mathrm{r}} \cdot \vec{\tau}=0$ and $\overrightarrow{\mathrm{F}} \cdot \vec{\tau} \neq 0$
Solution
Torque is an axial vector i.e., its direction is always perpendicular to the plane containing vectors $\overrightarrow{\mathrm{r}}$ and $\overrightarrow{\mathrm{F}}$.
$\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}$
Torque is perpendicular to both $\vec{r}$ and $\vec{F}$
$\therefore \quad \begin{aligned}
\vec{\tau} \cdot \overrightarrow{\mathrm{r}} & =0 \\
\overrightarrow{\mathrm{F}} \cdot \vec{\tau} & =0
\end{aligned}$
.