What is torque? Derive an expression for it in vector form.

Torque is a force that produces the turning or rotational motion in a body.
Torque is produced by a force perpendicular to the line joining the point about which the body rotates and the point of application of force.
Torque is a vector quantity and its direction is along the axis of rotation.
Consider a particle moving in KY plane along  the curve PQ under the influence of a force  
Let any instant t, the particle is at  and at instant t+dt, the particle is at  
Displacement undergone by the particle, 
Let F_x and F_y be the components of force , when the particle is at A. 
                      
The work done by force in displacing the body from A to B is,
 
From the above figure, we have

Substituting for dx and dy in equation(1), we get

From (1) and (2)
 
where, τ_z is the torque produced by force.
As the body revolves in XY plane, therefore torque by force will be along Z-axis.

Similarly, if the body moves in YZ plane, then torque along X-axis will be given by

τ = y^Fz ~ ^zFy                  ...(5)

If the body moves in XZ plane, then torque along Y-axis will be given by,

τ_y=zF_x-xF_z                      ...(6) 

If the body moves in space, then the torque produced by force is,
 
   
    

This is the required expression for torque.