Every quaternion z = a + bi + cj + dk can be viewed as a sum a + u of a real number a (called the "real part" of the quaternion) and a 3vector u = (b, c, d) = bi + cj + dk in R^{3} (called the "imaginary part"). In this view, quaternions are "mixed sums" of scalars and 3vectors and the quaternions i, j and k correspond to the unit vectors i, j and k.
Two such quaternions are added by adding the real parts and the imaginary parts separately:
This formula shows that two quaternions z and w commute, i.e. zw = wz, if and only if their imaginary parts are real multiples of each other (because in this case the vector product of these imaginary parts will commute).
It is well known that the vector product is related to rotation in space. The goal then is to find a formula which expresses rotation in 3d space using quaternion multiplication, similar to the formula for a rotation in 2d using complex multiplication:
The formula in 3d cannot be a simple multiplication with a quaternion, because multiplying a vector with a nontrivial quaternion yields a result with nonzero real part, and thus not a vector. Rotating a vector should yield a vector however.
It turns out that we can cancel the real part if we multiply by a quaternion from one side and with its inverse from the other side. Let z = a + u be a nonzero quaternion, and consider the function
Note that conjugation with z is the same as conjugation with rz for any real number r. We can thus restrict our attention to the quaternions of absolute value 1, the socalled unitquaternions. (The absolute value z of the quaternions z = a + v is defined as the square root of a^{2} + v^{2}. It is multiplicative: zw = z w.) Inverting unit quaternions is especially easy: if z = 1, then z^{1} = z^{*} (the conjugate z^{*} of the quaternion z = a + v is defined as z^{*} = a  v) and this makes our rotation formula even easier.
It turns out that the angle of rotation α is also easy to read off if we are dealing with a unit quaternion z = a + v: we have cos(α/2) = a. To summarize:
The composition of two rotations corresponds to quaternion multiplication: if the rotation f is represented by conjugation with the quaternion z and the rotation g is represented by conjugation with w, then the composition fog is represented by conjugation with zw.
If one wishes to rotate about an axis that doesn't pass through the origin, then one first translates the vectors into the origin, conjugates, and translates back.
Let us consider the rotation f around the axis u = i + j + k, with an angle of 120°, i.e. 2π/3 radians.
The length of u is √3, the half angle is π/3 with cosine 1/2 and sine (√3)/2. We are therefore dealing with a conjugation by the unit quaternion z = (1 + i + j + k)/2.
Concretely:
Quaternions vs. other representations of rotations
The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as a matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle. Both of these are much harder with matrices or Euler angles[?].
In computer games and other applications, one is often interested in "smooth rotations", meaning that the scene should slowly rotate and not in a single step. This can be accomplished by choosing a curve in the quaternions, with one endpoint being the identity transformation 1 and the other being the intended total rotation. This is more problematic with other representations of rotations.
When composing several rotations on a computer, rounding errors necessarily accumulate. A quaternion that's slightly off still represents a rotation  a matrix that's slightly off need not be orthogonal anymore and therefore need not represent a rotation at all. It is hard to turn such a matrix back into a proper orthogonal one.
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