Definition and relation to angular momentum:

A particle is located at position r relative to its axis of rotation. When a force F is applied to the particle, only the perpendicular component F? produces a torque. This torque t = r × F has magnitude t = r F? = r F sin? and is directed outward from the page.A force applied at a right angle to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm) is its torque. A force of three newtons applied two meters from the fulcrum, for example, exerts the same torque as a force of one newton applied six meters from the fulcrum.

The direction of the torque can be determined by using the right hand grip rule: if the fingers of the right hand curl in the direction of rotation and the thumb points along the axis of rotation, then the thumb also points in the direction of the torque.

More generally, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:
where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude t of the torque is given by
where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and ? is the angle between the position and force vectors.

Alternatively,
where F? is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque.

It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors. It points along the axis of rotation, and its direction is determined by the right-hand rule.The torque on a body determines the rate of change of its angular momentum, where L is the angular momentum vector and t is time. For rotation about a fixed axis,
where I is the moment of inertia and ? is the angular velocity. It follows that
where a is the angular acceleration of the body, measured in rad s-2.

Proof of the equivalence of definitions:

The definition of angular momentum for a single particle is:
where "×" indicates the vector cross product and p is the particle's linear momentum.

The time-derivative of this is:
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = mv,

we can see that:
But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma,

(Newton's 2nd law) we obtain:
And by definition, torque t = r × F.