Quaternions

Just like how complex numbers generalize real numbers, Quaternions further generalize complex numbers to 4 dimensions, with 3 “imaginary” components. A quaternion is defined as a 4-vector:

\[q := a + bi +cj + dk\]

The first dimension is called the real part and the latter three dimensions are called the vector part. Just like how complex numbers allow interaction between the real and imaginary components, quaternions similarly allow interactions via the following definitions:

• \[i^2 = j^2 = k^2 = -1\]
• \(i\) times \(j\) (i.e. \(ij\)) produces \(k\)
• \(j\) times \(k\) (i.e. \(jk\)) produces \(i\)
• \(k\) times \(i\) (i.e. \(ki\)) produces \(j\)
• The previous three rules, in the opposite order, require a multiplicative factor of \(-1\)

This seems hard to remember until you realize these rules function in a circle:

\[i \rightarrow j \rightarrow k \rightarrow i\]

Going in the reverse order requires multiplying by \(-1\).

Just as we can write all unit complex numbers as

\(z = \cos (\theta) + \sin (\theta) i\),

We can similarly write any quarternion in as

\[q = \cos(\theta) + \sin (\theta) (ai + bj + ck)\]

Properties

• Double cover: we can produce any desired rotation as \(q v \overbar{q} = (-q) v (- \overbar{q})\), meaning there are two ways to perform any rotation.