Quaternion

Definition

a ordered quadruple $(a, b, c, d)$ of real numbers Quaternions form a noncommutative division ring $H$ (introduced by Quaternions)

$q = a 1 + bi + c j + d k$

We can represent quaternions as

q = (a + i d b - i c - b - i c a - i d) = a (1001) + b (01 - 1 0) + c (0 - i - i 0) + d (i 0 0 - i)

with $det q = a^{2} + b^{2} + c^{2} + d^{2} = ∣ q ∣^{2}$

Warning

Not the only representation

Observation: Under this representation, the complex numbers are special quaternions with $c = d = 0$

The inverse $q^{- 1}$ of a quaternion $q$ is the matrix inverse of $q$

$i^{2} + j^{2} + k^{2} = - 1$ $ij = k, kj = i, ki = j, ji = - k$

$∣ q_{1} q_{2} ∣ = ∣ q_{1} ∣ ∣ q_{2} ∣$ by the multiplicative property of $det$ : $det (q_{1} q_{2}) = det (q_{1}) det (q_{2})$

q^{- 1} = \frac{1}{∣ q ∣ ^{2}} (a 1 - bi - c j - d k)

Let $u, v$ be pure imaginary quaternions (where $a = 0$ ) then $uv = - u \cdot v + u \times v$ (where $\times$ is the cross product)

Conjugate: $\overset{q}{ˉ} = a 1 - bi - c j - d k$

a rotation about axis $u$ through angle $θ$ corresponds to the quaternion pair $\pm q$ , where $q = cos (\frac{θ}{2}) + u sin (\frac{θ}{2})$