## 四元数 百科内容来自于：

### 性质特点

a、b、c、d是实数
i^2=j^2=k^2=-1
ij=k ji=-k jk=i kj=-i ki=j ik=-j
（a^2+b^2+c^2+d^2）的平方根 称为四元数的模.

### 例子

x = 3 + i
y = 5i + j - 2k

x + y = 3 + 6i + j - 2k
xy =( {3 + i} )( {5i + j - 2k} ) = 15i + 3j - 6k + 5i^2 + ij - 2ik
= 15i + 3j - 6k - 5 + k - 2j = - 5 + 15i + j - 5k

### 矩阵表示

$\begin a-di & -b+ci \\ b+ci & \;\; a+di \end$

$\begin\;\;a&-b&\;\;d&-c\\ \;\;b&\;\;a&-c&-d\\-d&\;\;c&\;\;a&-b\\ \;\;c&\;\;d&\;\;b&\;\;a\end$

### 综述

$q = a + \vec = a + bi + cj + dk$
$p = t + \vec = t + xi + yj + zk$

### 加法p + q

$p + q = a + t + \vec + \vec = (a + t) + (b + x)i + (c + y)j + (d + z)k$

### 乘法pq

$pq = at - \vec\cdot\vec + a\vec + t\vec + \vec\times\vec$
$pq = (at - bx - cy - dz) + (bt + ax + dy - cz)i + (ct + ay + bz - dx)j + (dt + za + cx - by)k \,$

$qp = at - \vec\cdot\vec + a\vec + t\vec - \vec\times\vec$

### 点积 p · q

$p \cdot q = at + \vec\cdot\vec = at + bx + cy + dz$

$p \cdot q = \frac{p^*q + q^*p}$

$p \cdot i = x$

### 外积Outer(p,q)

$\operatorname(p,q) = \frac{p^*q - q^*p}$
$\operatorname(p,q) = a\vec - t\vec - \vec\times\vec$
$\operatorname(p,q) = (ax - tb - cz + dy)i + (ay - tc - dx + bz)j + (az - td - by + xc)k$

### 偶积

$\operatorname(p,q) = \frac{pq + qp}$
$\operatorname(p,q) = at - \vec\cdot\vec + a\vec + t\vec$
$\operatorname(p,q) = (at - bx - cy - dz) + (ax + tb)i + (ay + tc)j + (az + td)k$

### 叉积：p × q

$p \times q = \frac{pq - qp}$
$p \times q = \vec\times\vec$
$p \times q = (cz - dy)i + (dx - bz)j + (by - xc)k$

### 转置

$p^ = \frac{p^*}{p\cdot p}$

### 标量部

$1\cdot p = \frac{p + p^*} = a$

### 向量部

$\operatorname（1,p) = \frac{p - p^*} = \vec = bi + cj + dk$

### 模：|p|

$|p| = \sqrt{p \cdot p} = \sqrt{p^*p} = \sqrt{a^2 + b^2 + c^2 + d^2}$

### 符号数：sgn(p)

$\sgn(p) = \frac{|p|}$

### 幅角：arg(p)

$\arg(p) = \arccos\left(\frac{\operatorname(p)}{|p|}\right)$
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