An external force on a particle constitutes a torque with respect to a point. Only condition is that the point, about which torque is defined or measured, does not lie on the line of force, in which case torque is zero.
Torque on a particle 

We can make note of the fact that it is convenient to construct reference system in such a manner so that the point (about which torque is measured) coincides with the origin. In that case, the vector ( r) denoting the position of the particle with respect to point, becomes the position vector, which is measured from the origin of reference system.
Magnitude of torque
With the reference of origin for measuring torque, we can find the magnitude of torque, using any of the following relations given below. Here, we have purposely considered force in xy  plane for illustration and visualization purpose as it provides clear directional relationship of torque " τ " with the operand vectors " r " and " F ".
Torque on a particle  


1: Torque in terms of angle enclosed
2: Torque in terms of force perpendicular to position vector
3: Torque in terms of moment arm
Torque on a particle 

Example 1
Problem : A projectile of mass "m" is projected with a speed "v" at an angle "θ" with the horizontal. Calculate the torque on the particle at the maximum height in relation to the point of projection.
Solution : The magnitude of the torque is given by :
Projectile motion 

where
Now, the force on the projectile of mass "m" is due to the force of gravity :
Putting these expressions of moment arm and force in the expression of torque, we have :
Application of right hand rule indicates that torque is clockwise and is directed in to the page.
Direction of torque
The determination of torque's direction is relatively easier than that of angular velocity. The reason is simple. The torque itself is equal to vector product of two vectors, unlike angular velocity which is one of the two operands of the vector product. Clearly, if we know the directions of two operands here, the direction of torque can easily be interpreted.
By the definition of vector product, the torque is perpendicular to the plane formed by the position vector and force. Besides, it is also perpendicular to each of the two vectors individually. However, the vector relation by itself does not tell which side of the plane formed by operands is the direction of torque. In order to decide the orientation of the torque, we employ right hand vector product rule.
For this, we need to shift one of the operand vectors such that their tails meet at a point. It is convenient to shift the force vector, because application of right hand vector multiplication rule at the point (origin of the coordinate system) gives us the sense of angular direction of the torque.
Vector product  


In the figure, we shift the second vector (F) so that tails of two operand vectors meet at the point (origin) about which torque is calculated. The two operand vectors define a plane ("xy"  plane in the figure). The torque (τ) is, then, acting along a line perpendicular to this plane and passing through the meeting point (origin).
The orientation of the torque vector in either of two direction is determined by applying right hand rule. For this, we sweep the closed fingers of the right hand from position vector (first vector) to force vector (second vector). The outstretched thumb, then, indicates the orientation of torque. In the case shown, the direction of torque is positive zdirection.
While interpreting the vector product, we must be careful about the sequence of operand. The vector product "