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We know that torque is given by $$\vec{\tau} = \vec{r} \times \vec{F}.$$ Its direction is given by the right-hand rule which says that torque acts perpendicular to the plane where force applied and position vector do exist. But if the direction of the torque is upward, why does the body rotate in the plane perpendicular to the direction?

For clarification, if one applies a force $\vec{F}$ on the door at a point which is $\vec{r}$ away from the hinge, the torque $\vec{\tau}$ is given by the above relation. But in which direction does it act? It acts perpendicular to the force i.e. along the length of the door. But why does the door rotate perpendicular to the direction? What does the direction of the torque imply if the body doesn't move in that direction?

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Now, torque τ⃗ is given by the above relation. But in which direction does it act?

Torque, like angular momentum is a pseudovector and not a vector. It is a conventional way of showing the direction (anti/clock -wise) ot the rotation. It is devised in such a way that you can apply the right-hand rule.

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