Question
Let $\vec{\text{F}}$ be a force acting on a particle having position vector $\vec{\text{r}}.$ Let $\vec{\tau}$ be the torque of this force about the origin, then:
  1. $\vec{\text{r}}.\vec{\text{F}}=0$ and $\vec{\text{F}}.\vec{\tau}=0$
  2. $\vec{\text{r}}.\vec{\tau}=0$ but $\vec{\text{F}}.\vec{\tau}\neq0$
  3. $\vec{\text{r}}.\vec{\tau}\neq0$ but $\vec{\text{F}}.\vec{\tau}=0$
  4. $\vec{\text{r}}.\vec{\tau}\neq0$ and $\vec{\text{F}}.\vec{\tau}\neq0$

Answer

  1. $\vec{\text{r}}.\vec{\text{F}}=0$ and $\overrightarrow{\text{F}}.\overrightarrow{\tau}=0$
Explanation:
$\vec{\tau}=\vec{\text{r}}\times\vec{\text{F}}$
Thus, $\vec{\tau}$ is perpendicular to $\vec{\text{r}}$ and $\vec{\text{F}}$
Therefore, we have:
$\vec{\text{r}}.\vec{\tau}=0$ and $\vec{\text{F}}.\vec{\tau}=0$

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