Quaternion for beginner QUATERNION ROTATIONI have 2 points in 3d space ,point A= <5,3,6> and B=<8,2,3>
I want to rotate "point B" by 30 degree from point A.
how do I solve this question using quaternion.
Plz, explain the steps....
 A: You can't actually rotate one point about another point, thats not what quaternions do. However, you can rotate a point (P1 in the picture or B in your question) about a rotation axis (which is your A) as shown:

1. Constructing the Rotation Quaternion
If you want to achieve a 30 degree rotation, you first need to construct the quaternion that applies the rotation to your point. To do this, first find a unit vector in the direction of A, which is given by:
$$ \hat a = {\vec A \over \left | A \right| } = a_x \hat i + a_y \hat j + a_z \hat k$$
Assuming you now have a unit vector, and want to achieve a 30 degree rotation, you can set $\theta = 30\deg$ and find the quaternion that applies this rotation with:
$$ q = \cos \left( {\theta \over 2} \right) + \hat a\sin \left( {\theta \over 2} \right)$$
At this stage, you will have a single quaternion which has a real component ($\cos \left( {\theta \over 2} \right)$) and an imaginary component, which is the three coefficients from your unit vector multiplied by ($\sin \left( {\theta \over 2} \right)$).
2. Applying the Rotation
Given that the point you are rotating is $B = b_x \hat i + b_y \hat j + b_z \hat k$, you can imagine this as the purely imaginary quaternion: 
$$p =  b_x i + b_y  j + b_zk $$
Now you apply the conjugation formula (which rotates your point p), to this point to get your new point, which we can call $p_\text{new}$, which is given by:
$$p_\text{new} = qpq^{-1}$$
Where $q^{-1} = \cos \left( {\theta \over 2} \right) - \hat a\sin \left( {\theta \over 2} \right)$. Then the complex part of $p_\text{new}$ is your rotated point. Majic! :P
A: if i want to want to rotate vector [3,5,-2] 90 degrees about X axis. Our angle x is 90 and axis v is [1,0,0] (X axis). 
The quaternion q is:
q = cos(45) + [1,0,0]*sin(45) = 0.707 + 0.707*i
It turns out, that:
1/q = 0.707 - 0.707*i
Vector w, as a quaternion, is 3i + 5j - 2k. Now we have to calculate:
(0.707+0.707*i)(3i+5j-2k)(0.707-0.707*i) = (after many calculations) = 3i + 2j + 5k
So, the rotated vector is [3,2,5].
