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 $$\operatorname{ V } \left( q \right)$$
$a_1+b_1\mathbf i + c_1\mathbf j + d_1\mathbf k) + (a_2 + b_2\mathbf i + c_2\mathbf j + d_2\mathbf k) = (a_1 + a_2) + (b_1 + b_2)\mathbf i + (c_1 + c_2)\mathbf j + (d_1 + d_2) \mathbf k,$
$\lambda(a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k) = \lambda a + (\lambda b)\,\mathbf i + (\lambda c)\,\mathbf j + (\lambda d)\,\mathbf k.$
Given the quaternion $$q = a + b \mathbf{ i }+ c \mathbf{ j } + d \mathbf{ k }$$, it's conjugate is given by $\overline{ q } := a - b \mathbf{ i }- c \mathbf{ j } - d \mathbf{ k }$
A pure quaternion $$a + b \mathbf{ i }+ c \mathbf{ j } + d \mathbf{ k }$$ is one such that $$a = 0$$