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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
A
$\vec{r}\cdot\vec{\tau} = 0$ and $\vec{F}\cdot\vec{\tau} \neq 0$
B
$\vec{r}\cdot\vec{\tau} \neq 0$ and $\vec{F}\cdot\vec{\tau} = 0$
C
$\vec{r}\cdot\vec{\tau} > 0$ and $\vec{F}\cdot\vec{\tau} < 0$
D
$\vec{r}\cdot\vec{\tau} = 0$ and $\vec{F}\cdot\vec{\tau} = 0$
Detailed Solution
Torque is defined as the cross product $\vec{\tau} = \vec{r}\times\vec{F}$
The cross product of two vectors is perpendicular to both of them, so $\vec{\tau}$ is perpendicular to $\vec{r}$ and also to $\vec{F}$.
The dot product of two perpendicular vectors is zero.
$\vec{r}\cdot\vec{\tau} = r\tau\cos90^\circ = 0$
$\vec{F}\cdot\vec{\tau} = F\tau\cos90^\circ = 0$
Hence $\vec{r}\cdot\vec{\tau} = 0$ and $\vec{F}\cdot\vec{\tau} = 0$.
The cross product of two vectors is perpendicular to both of them, so $\vec{\tau}$ is perpendicular to $\vec{r}$ and also to $\vec{F}$.
The dot product of two perpendicular vectors is zero.
$\vec{r}\cdot\vec{\tau} = r\tau\cos90^\circ = 0$
$\vec{F}\cdot\vec{\tau} = F\tau\cos90^\circ = 0$
Hence $\vec{r}\cdot\vec{\tau} = 0$ and $\vec{F}\cdot\vec{\tau} = 0$.