Addition of quaternions

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Two quaternions can easily be added by adding their components.

Assume two quaternions:

$\begin{align} q &= q_0 + q_1i + q_2j + q_3k \\ p &= p_0 + p_1i + p_2j + p_3k \end{align}$

The quaternions $q$ and $p$ can just be replaced by these equations in their addition $q+p$ and then be summarized as the individual four components:

$\begin{align} q+p &= q_0 + q_1i + q_2j + q_3k + p_0 + p_1i + p_2j + p_3k \\ &= (q_0+p_0) + (q_1+p_1)i + (q_2+p_2)j + (q_3+p_3)k \end{align}$

So the addition in vector notation can be written as:

$\begin{align} q+p &= q_0+p_0 \oplus \vec{\mathbf{q}}+\vec{\mathbf{p}} \end{align}$

Obviously addition of two quaternions is commutative and associative:

$\begin{align} q+p&=p+q \\ (q+p)+r&=q+(p+r) \end{align}$