Let F be the force acting on a particle having position vector r, and τ be the torque of this force about the origin. Then (a) r·τ = 0 and F·τ ≠ 0 (b) r·τ ≠ 0 and F·τ = 0 (c) r·τ ≠ 0 and F·τ ≠ 0 (d) r·τ = 0 and F·τ = 0

Answer: (d). τ = r is perpendicular to both r and F. So both dot products are zero: r·τ = 0, F·τ = 0. So the answer is (d).

Let F be the force acting on a particle having position vector r, and… | JEE Main 2003 (AIEEE)

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Torque, Couple & Equilibrium of Rigid Bodies

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

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Answer: (d)

$\vec{\tau} = \vec{r}\times\vec{F}$ is perpendicular to both $\vec{r}$ and $\vec{F}$ .
So both dot products are zero: $\vec{r}\cdot\vec{\tau} = 0$ , $\vec{F}\cdot\vec{\tau} = 0$ .
So the answer is (d).