Let $\vec{F}$ be the force acting on a particle having position vector $\vec{r}$, and $\vec{\tau}$ be the torque of this force about the origin. Then
Answer: (D) $\vec{r}\cdot\vec{\tau} = 0$ and $\vec{F}\cdot\vec{\tau} = 0$
Concept. Torque about the origin is $\vec\tau=\vec r\times\vec F$. A cross product is always perpendicular to both of its factors, and perpendicular vectors have zero dot product.
Step 1. Direction of $\vec\tau$. By definition of the cross product, $\vec\tau$ is perpendicular to the plane containing $\vec r$ and $\vec F$.
Step 2. Dot products. Since $\vec\tau\perp\vec r$ and $\vec\tau\perp\vec F$:
$$\vec r\cdot\vec\tau=\vec r\cdot(\vec r\times\vec F)=0,\qquad \vec F\cdot\vec\tau=\vec F\cdot(\vec r\times\vec F)=0$$
(Each is a triple product with a repeated vector, which is zero.)
Answer. Option D: both dot products are zero.
Solution by Sreeraj P, M.Sc Physics
