Quaternion, dot product and cross product, number theory

This article will introduce the connection between dot product and cross product by considering quaternion.

An element of $\mathbb{Q}$$\mathbb Q$ can be represented as $a+bi+cj+dk,a,b,c,d\in \mathbb{R}$$a+bi+cj+dk,a,b,c,d\in\mathbb R$

$i^2=j^2=k^2=ijk=-1$$i^2=j^2=k^2=ijk=-1$, $ij=k=-ji,jk=i=-kj,ki=j=-ik$$ij=k=-ji,jk=i=-kj,ki=j=-ik$

And consider $u,v\in {\mathbb{R}}^3,u=u_{1}i+u_{2}j+u_{3}k,v=v_{1}i+v_{2}j+v_{3}k$$u,v\in\mathbb R^3,u=u_1i+u_2j+u_3k,v=v_1i+v_2j+v_3k$

And $uv=-⟨u,v⟩+u\times v$$uv=-\langle u,v\rangle+u\times v$

That is why one is $\cos \theta$$\cos\theta$, another is $\sin \theta$$\sin\theta$

And we can represent $a+bi+cj+dk,a,b,c,d\in \mathbb{R}$$a+bi+cj+dk,a,b,c,d\in\mathbb R$ as a matrix.

$\begin{array}{c}T=\left(\begin{array}{cc}a+di& -b-ci\\ b-ci& a-di\end{array}\right)\end{array}$$T=\begin{pmatrix} a+di & -b-ci \\ b-ci & a-di \end{pmatrix}$

Another interesting thing we can prove by $\mathbb{Q}$$\Q$ is

$\mathrm{Proposition}.$$\mathrm{Proposition.}$

for $s,t\in A:=\{a^2+b^2+c^2+d^2,a,b,c,d\in \mathbb{Z}\},st\in A$$s,t\in A:=\{a^2+b^2+c^2+d^2,a,b,c,d\in\mathbb Z\},st\in A$

$\mathrm{Proof}.$$\mathrm{Proof.}$

define the norm of $\mathbb{Q}$$\Q$ can be written as $q\bar{q}=(a+bi+cj+dk)(a-bi-cj-dk)=a^2+b^2+c^2+d^2$$q\overline q=(a+bi+cj+dk)(a-bi-cj-dk)=a^2+b^2+c^2+d^2$，

Actually, it is $\begin{array}{c}det\left(\begin{array}{cc}a+di& -b-ci\\ b-ci& a-di\end{array}\right)\end{array}$$\det\begin{pmatrix} a+di & -b-ci \\ b-ci & a-di \end{pmatrix}$

According to $det(AB)=det(A)det(B)$$\det(AB)=\det(A)\det( B)$

We prove that for $s,t\in A:=\{a^2+b^2+c^2+d^2,a,b,c,d\in \mathbb{Z}\},st\in A$$s,t\in A:=\{a^2+b^2+c^2+d^2,a,b,c,d\in\mathbb Z\},st\in A$