Rotation formalisms in three dimensions I'm little bit confused. The Rotations are described by various means
Direction Cosines Matrix (DCM);
Euler Angles;
Euler Axis/Angle;
Quaternion.
What is the difference between them. How I can convert the orientation from one representation to another.
My question here: I have a the orientation of an object in terms of direction cosine vector. 
See the following figure; direction cosine - unit vector
Note that the vector r makes angle alpha with x-axis, beta with y-axis and gamma with z-axis.
I want to represent this orientation in tearms of Quaternion. Is that possible? if yes, How we can represent this angles in terms of Quaternion (Axis,Angle,Matrix).
Thanks for the Help!.
 A: The wiki article provides all the conversions you could ask for. Specifically you may want this section.
The formalism (or system) you eventually choose depends on your application. In theory, using any or all of those systems is possible.
Rather than rehashing all of that stuff here, you might want to spend time to pick out the one that sounds right to you, and then ask some targeted question about it.

Update: The request was clarified to changing from a direction cosine to a quaternion. The quaternion needs two pieces of data: the components of the unit vector $u$ on the axis of rotation, and the angle of rotation $\theta$ (measured in a right-hand manner in the plane normal to $u$.)
The three angles in your picture will give you $u$, and I assume you want to specify $\theta$ later. Finding the components of the vector in the picture is easy: the projections onto the axes are just $\cos(\alpha)$, $\cos(\beta)$, and $\cos(\gamma)$, so $u=i\cos(\alpha)+j\cos(\beta)+k\cos(\gamma)$.
In the scheme of quaternion rotations, the matching quaternion is $q=\cos(\theta/2)+u\sin(\theta/2)$.
