What does torque measure in terms of producing rotational motion?

$A$ force $\vec{F}$ acts on a particle with position vector $\vec{r}$. The torque $\vec{\tau}$ about the origin due to this force is given by $\vec{\tau} = \vec{r} \times \vec{F}$. Which of the following is true?

Give the formula for the moment of a couple.

When a force of $6.0 \, N$ is exerted at $30^{\circ}$ to a wrench at a distance of $8 \, cm$ from the nut,it is just able to loosen the nut. What force $F$ would be sufficient to loosen it,if it acts perpendicularly to the wrench at $16 \, cm$ from the nut (in $, N$)?

$A$ door is hinged at one end and is free to rotate about a vertical axis (figure). Does its weight cause any torque about this axis? Give a reason for your answer.

Three forces $F, 2F$ and $3F$ act on a rod $AB$ which is pivoted at $A$. The anticlockwise moments of forces $F, 2F$ and $3F$ about the pivot $A$ are respectively: