The Quaternions
The collection of quaternions are: \[ \mathbb{ H } := \left\{ a + b \mathbf{ i } + c \mathbf{ j }+ d \mathbf{ k } \right\} \] where \( a, b, c, d \in \mathbb{ R } \) and \( \mathbf{ i } ^ 2 = \mathbf{ j } ^ 2 = \mathbf{ k } ^ 2 = \mathbf{ ijk }= -1 \)
Scalar Part of the Quaternion
Given a quaternion \(q = a + bi + cj + dk\), we say that \(a\) is the scalar part of the quaternion (or real part), and is denoted by \(\operatorname{ Re } \left( q \right) \)
Vector Part of the Quaternion
Given a quaternion \(q = a + bi + cj + dk\), we say that \(bi + cj + dk\) is the vector part of the quaternion , and is denoted by \(\operatorname{ V } \left( q \right) \)
Quaternion Addition
\[ a_1+b_1\mathbf i + c_1\mathbf j + d_1\mathbf k) + (a_2 + b_2\mathbf i + c_2\mathbf j + d_2\mathbf k) = (a_1 + a_2) + (b_1 + b_2)\mathbf i + (c_1 + c_2)\mathbf j + (d_1 + d_2) \mathbf k, \]
Scalar Quaternion Multiplication
\[ \lambda(a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k) = \lambda a + (\lambda b)\,\mathbf i + (\lambda c)\,\mathbf j + (\lambda d)\,\mathbf k. \]
Conjugate Quaternion
Given the quaternion \( q = a + b \mathbf{ i }+ c \mathbf{ j } + d \mathbf{ k } \), it's conjugate is given by \[ \overline{ q } := a - b \mathbf{ i }- c \mathbf{ j } - d \mathbf{ k } \]
Pure Quaternion
A pure quaternion \( a + b \mathbf{ i }+ c \mathbf{ j } + d \mathbf{ k } \) is one such that \( a = 0 \)