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Show that if $q_1 = s_1 + v_1$ and $q_2 = s_2 + v_2$ are two quaternions with scalar parts $s_1, s_2, $ and vector parts $v_1, v_2$, then their product is the quaternion with the following simplification rules:

$i^2=j^2=k^2=-1$,

$ij=k=-ij$,

$jk=i=-kj$,

$ki=j=-ik$

$$q_1 q_2 = (s_1s_2 - v_1 \cdot v_2) + (s_1v_2 + s_2v_1 + v_1 \times v_2)$$

I figured I would use distributive multiplication on $q_1$ and $q_2$ and so far I have

$q_1 = (a_1 + b_1i + c_1j + d_1k)$ and $q_2 = (a_2 + b_2i + c_2j + d_2k)$

$\begin{align} q_1q_2 &= a_1a_2 + a_1b_2i + a_1c_2j + ad_2k + \\ & a_2b_1i + b_1b_2i^2 + b_1c_2ij + b_1d_2ki + \\ & c_1a_2j + c_1b_2ji + c_1c_2j^2 + c_1d_2ik + \\ & d_1a_2k + d_1b_2ki + d_1c_2kj + d_1d_2k^2 \end{align}$

However when I try to simplify further from this point, I feel like I am making copious amounts of mistakes and am struggling to stay organized. Is there a better way to go about doing this or at least organizing my work to get to the end?

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Instead of expanding everything at once, it might help you keep things organized to delay expanding the vector part. Writing $q$ as the sum $s + \mathbf v$ as suggested, it shouldn’t be hard to convince yourself that quaternion multiplication distributes over $+$. So, $$(s_1+\mathbf v_1)(s_2+\mathbf v_2) = s_1s_2 + s_1\mathbf v_2 + s_2\mathbf v_1 + \mathbf v_1\mathbf v_2.$$ The first term $s_1s_2$ is a scalar while $s_1\mathbf v_2$ and $s_2\mathbf v_1$ are vectors, so after grouping them the expanded product looks like $$(s_1s_2 + \dots)+(s_1\mathbf v_2+s_2\mathbf v_1+\dots).$$ You’re then left with working out the quaternion product $\mathbf v_1\mathbf v_2$ of two pure vectors. Fully expanded, this product consists of nine terms instead of your original sixteen, so should be much easier to manage.

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I like the approach that amd has used above, and suggest you give it a look.

However, from where you are.

$\begin{align} q_1q_2 &= a_1a_2 + a_1b_2i + a_1c_2j + ad_2k + \\ & a_2b_1i + b_1b_2i^2 + b_1c_2ij + b_1d_2ki + \\ & c_1a_2j + c_1b_2ji + c_1c_2j^2 + c_1d_2jk + \\ & d_1a_2k + d_1b_2ki + d_1c_2kj + d_1d_2k^2 \end{align}$

replace $i^2, j^2, k^2$ with $-1$ and $ij = k, ji = -k,$ etc. we get

$\begin{align} q_1q_2 &= a_1a_2 + a_1b_2i + a_1c_2j + ad_2k + \\ & a_2b_1i - b_1b_2 + b_1c_2k - b_1d_2j + \\ & c_1a_2j - c_1b_2k - c_1c_2 + c_1d_2i + \\ & d_1a_2k + d_1b_2j + d_1c_2kj - d_1d_2 \end{align}$

Collect the real parts and the $i,j, k$ parts

$\begin{align} q_1q_2 &= a_1a_2 - b_1b_2 - c_1c_2-d_1d_2+ \\ &(a_1b_2 + a2b1 + c_1d_2 - c_2d_1)i +\\ &(a_1c_2 + a_2c_1 -b_1d_2+b_2d1)j+\\ &(a_1d_2 + a2d_1 + b_1c_2 - b_2c_1)k \end{align}$

$\begin{align} q_1q_2 &= a_1a_2 - (b_1b_2 + c_1c_2+d_1d_2)+ \\ &a_1(b_2i+c_2j+d_2j) + a_2(b_1i + c_1j+d_1k)+\\ &(c_1d_2-c_2d_1)i + (b_2d_1 - b_1d_2)j + (b_1c_2 - b_2c_1)k \end{align}$

$(b_1b_2 + c_1c_2+d_1d_2) = v_1\cdot v_2$ and $(c_1d_2-c_2d_1)i + (b_2d_1 - b_1d_2)j + (b_1c_2 - b_2c_1)k = v_1\times v_2$

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