First let's recall the conditions for particle equilibrium.
According to Newton’s first law, an object will have a linear acceleration of zero when there is no unbalanced force acting on it. 
In other words, the resultant force vector acting on the object must be zero. 
Since particle is an idealized object with no size or shape, 
and is only represented by a dot in space, the forces acting on the particle will be concurrent 
and for a 2D problem, the vector equation 
can be written as two scalar equations 
that the resultant forces in the x and y direction must equal to zero respectively, 
And for a 3D problem, the vector equation becomes three scalar equations,
that the resultant forces in the x, y and z direction must all be zero. 
Therefore, for a 2D problem, based on one free body diagram, we can write two scalar equations
and then solve for two unknowns. For a 3D problem, based on one free body diagram, 
we can write three scalar equations and that will enable us to solve for three unknowns. 
However, a rigid body has shape and size and it is not necessarily static even if the resultant 
force acting on it is indeed zero. For example, if you recall the moment of a couple:
the two forces acting on this wheel are indeed in equilibrium, 
but this only means that they don’t cause translational motion. We already learned that these two forces make a couple moment, F d, 
which is a free moment that causes rotational effect of this wheel. 
Therefore, for a rigid body to be static, it is not enough to only have no unbalanced force, but  
the resultant moment summarized about any arbitrary point must be zero as well.
Otherwise the object will rotate.
Therefore, for a rigid body that is subjected to multiple forces and couple moments, the first condition for equilibrium is the same as 
particle equilibrium, that the resultant external force acting on the body must be zero. 
Then, the resultant moment summarized about any arbitrary point O, 
must also be zero. This resultant moment includes both the total moment caused by the forces, and 
the total couple moments. Note that both of these equations are vector equations. 
As a summary, for rigid body equilibrium, we can have two vector equations, one for force and one for moment. For a 2D problem, based on
one free body diagram, we can write a maximum of three independent scalar equations and then solve for three unknowns.
Note that generally we will write two force equations along x and y directions and one  moment
equation about an arbitrary point in the 2D plane. But you can write three alternative equations, for example,
one force equation and two moment equations about two different points A and B in the x-y plane, or even, three moment equations about 
three different points A, B and C in the x-y plane. Depending on the specific problem you have,
you choose what equations are the most convenient and helpful to write. 
However, please remember for one 2D free body diagram you can only have a maximum of three independent equations. If you write more,  
the excess equations can be derived from the others and therefore will not help you solve for more unknowns. 
For a 3D problem, based on one free body diagram, we can write a maximum of six independent scalar equations and therefore can 
solve for a maximum of six unknowns. These are the three force equilibrium equations along the x, y and z direction respectively, 
and the three moment equilibrium equations about the x, y and z axis respectively. 
Here are two examples of rigid body equilibrium problems we might find. 
Normally the applied loadings are known, and we will need to use the equilibrium equations to find the unknown support reactions. 
Don’t forget, the support reactions are also external force or moment acting on the body. 
To do that, we will need to first identify what kinds of reactions are associated with different types of supports. 
For that, please watch the other videos on the topic of supports. 
