Why is moment of a force calculated by cross product? Why the moment of a force is calculated by the cross product of two vectors,how it is related ? And what does it means ,when I said that moment of a force about a point is for example 6 N.m ? I want simple clear explanation.
 A: The word torque derives from the italian "torcere" which means "to twist". Torque or moment of a force is actually a measure of how effective that force is in twisting an object about an axis.
To motivate its definition let us say we are interested in twisting a screw $O$ by applying a force $F$ to a wrench of length $r$.

The effectiveness of that twisting is proportional to the applied force as well as the length of the tool. Moreover, if the angle between the force and the wrench is zero the effectiveness vanishes whereas it is maximum when the angle is 90 degrees. Hence it is reasonable to define that effectiveness as
$$\tau=Fr\sin\theta.$$
Actually we also want to know the axis of rotation as well as if that force tends to twist the screw in on sense or the other so we can assign to the effectiveness some direction and sense. The sense is conventioned by the right hand rule. Therefore, toghether with the above magnitude we can define the torque as
$$\vec\tau=\vec r\times\vec F,$$
where $\vec r$ is the vector from the pivot (the screw) to the point where the force is applied. Note that $|\vec\tau|=rF\sin\theta$.
If someone tells me that a torque is $9\, Nm$ I can think of infinitely many possibilities since that is not enough information. For example, There is a wrench of $9\, m$ and a force of $1\, N$ making an angle of $90$ degrees. Or it may be a wrench of $1\, m$ with a force of $18\, N$ and angle of $60$ degrees. Another type of information (more useful for physicists) that can be obtained from torque comes from Newton's second law for rotation. In general it says that the rate of change of angular momentum equals the total external torque,
$$\frac{d\vec L}{dt}=\vec \tau.$$
For many interesting cases it cal also be written as
$$\tau=I\alpha,$$
where $I$ is the moment of inertia of the body and $\alpha$ its angular acceleration. So a $9\, Nm$ torque is able for example to impress an angular acceleration of $9\, rad/s^2$ to a body with moment of inertia of $1\,kg m^2$. Yet there are too many possibilities of bodies and accelerations.
