Imagine you are using a wrench to loosen a nut. The idea is to turn the nut by applying a force to the end of the wrench. If you push or pull the wrench directly toward or away from the nut, it should be obvious from experience that no progress will be made in turning the nut. The important thing is the component of force perpendicular to the wrench. It is this component of force which will cause the nut to turn. For example see the following picture.
In the picture a force, F is applied at the end of a wrench represented by the position vector R and the angle between these two is θ. Then the tendency to turn will be

This is also called the moment of the force, F. That way, if there are several forces acting at several points the total torque can be obtained by simply adding up the torques associated with the different forces and positions.
Example 3.5.2 Suppose R_{1} = 2i − j+3k,R_{2} = i+2j−6k meters and at the points determined by these vectors there are forces, F_{1} = i − j+2k and F_{2} = i − 5j + k Newtons respectively. Find the total torque about the origin produced by these forces acting at the given points.
It is necessary to take R_{1} × F_{1} + R_{2} × F_{2}. Thus the total torque equals

Example 3.5.3 Find if possible a single force vector F which if applied at the point i + j + k will produce the same torque as the above two forces acting at the given points.
This is fairly routine. The problem is to find F = F_{1}i + F_{2}j + F_{3}k which produces the above torque vector. Therefore,

which reduces to

However, there is no solution to these three equations. (Why?) Therefore no single force acting at the point i + j + k will produce the given torque.