向量运算

基本运算

加法

点积

三维向量的点积定义如下：

$$ \overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}=u_{x} v_{x}+u_{y} v_{y}+u_{z} v_{z}=|\overrightarrow{\mathbf{u}} || \overrightarrow{\mathbf{v}} | \cos (\overrightarrow{\mathbf{u}}, \overrightarrow{\mathbf{v}}) $$

叉积

三维向量的叉积定义如下：

$$ \overrightarrow{\mathbf{w}}=\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}=\left[\begin{array}{ccc}{\overrightarrow{\mathbf{i}}} & {\overrightarrow{\mathbf{j}}} & {\overrightarrow{\mathbf{k}}} \ {u_{x}} & {u_{y}} & {u_{z}} \ {v_{x}} & {v_{y}} & {v_{z}}\end{array}\right] $$

其中$\overrightarrow{\mathbf{i}}, \overrightarrow{\mathbf{j}}, \overrightarrow{\mathbf{k}}$分别为$x,y,z$轴的单位向量：

$$ \overrightarrow{\mathbf{u}}=u_{x} \overrightarrow{\mathbf{i}}+u_{y} \overrightarrow{\mathbf{j}}+u_{z} \overrightarrow{\mathbf{k}}, \quad \overrightarrow{\mathbf{v}}=v_{x} \overrightarrow{\mathbf{i}}+v_{y} \overrightarrow{\mathbf{j}}+v_{z} \overrightarrow{\mathbf{k}} $$

$\overrightarrow{\mathbf{u}}$和$\overrightarrow{\mathbf{v}}$的叉积垂直于$\overrightarrow{\mathbf{u}}, \overrightarrow{\mathbf{v}}$构成的平面，其方向符合右手规则。叉积的模等于$\overrightarrow{\mathbf{u}}, \overrightarrow{\mathbf{v}}$构成的平行四边形的面积，且符合如下的条件：

$$ \begin{array}{l}{\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}=-\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{u}}} \ {\overrightarrow{\mathbf{u}} \times(\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{w}})=(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{w}}) \overrightarrow{\mathbf{v}}-(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}) \overrightarrow{\mathbf{w}}}\end{array} $$

混合积

$$ \begin{array}{rl}{[\overrightarrow{\mathbf{u}} \overrightarrow{\mathbf{v}}} & {\overrightarrow{\mathbf{w}} ]=(\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}) \cdot \overrightarrow{\mathbf{w}}=\overrightarrow{\mathbf{u}} \cdot(\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{w}})}

= \left|\begin{array}{lll}{u_{x}} & {u_{y}} & {u_{z}} \ {v_{x}} & {v_{y}} & {v_{z}} \ {w_{x}} & {w_{y}} & {w_{z}}\end{array}\right|

= \left|\begin{array}{lll}{u_{x}} & {v_{x}} & {w_{x}} \ {u_{y}} & {v_{y}} & {w_{y}} \ {u_{z}} & {v_{z}} & {w_{z}}\end{array}\right|

\end{array} $$

其物理意义为：以$\overrightarrow{\mathbf{u}}, \overrightarrow{\mathbf{v}}, \overrightarrow{\mathbf{w}}$为三个棱边所围成的平行六面体的体积。当$\overrightarrow{\mathbf{u}}, \overrightarrow{\mathbf{v}}, \overrightarrow{\mathbf{w}}$

并矢

给定两个向量$\overrightarrow{\mathbf{x}}=\left(x_{1}, x_{2}, \cdots, x_{n}\right)^{T}, \overrightarrow{\mathbf{y}}=\left(y_{1}, y_{2}, \cdots, y_{m}\right)^{T}$，则向量的并矢记作：

$$ \overrightarrow{\mathbf{x}} \overrightarrow{\mathbf{y}}=\left[\begin{array}{cccc}{x_{1} y_{1}} & {x_{1} y_{2}} & {\cdots} & {x_{1} y_{m}} \ {x_{2} y_{1}} & {x_{2} y_{2}} & {\cdots} & {x_{2} y_{m}} \ {\vdots} & {\vdots} & {\ddots} & {\vdots} \ {x_{n} y_{1}} & {x_{n} y_{2}} & {\cdots} & {x_{n} y_{m}}\end{array}\right] $$

也记作$\overrightarrow{\mathbf{x}} \otimes \overrightarrow{\mathbf{y}}$或者$\overrightarrow{\mathrm{x}} \overrightarrow{\mathbf{y}}^{T}$。

线性相关

一组向量$\overrightarrow{\mathbf{v}}{1}, \overrightarrow{\mathbf{v}}{2}, \cdots, \overrightarrow{\mathbf{v}}{n}$如果是线性相关的，那么值存在一组不全为零的实数，$a{1}, a_{2}, \cdots, a_{n}$，使得$\sum_{i=1}^{n} a_{i} \overrightarrow{\mathbf{v}}_{i}=\overrightarrow{\mathbf{0}}$。

反之，一组向量$\overrightarrow{\mathbf{v}}{1}, \overrightarrow{\mathbf{v}}{2}, \cdots, \overrightarrow{\mathbf{v}}{n}$如果是线性无关的，当且仅当$a{i}=0, i=1,2, \cdots, n$才有$\sum_{i=1}^{n} a_{i} \overrightarrow{\mathbf{v}}_{i}=\overrightarrow{\mathbf{0}}$。

向量性质

维数

一个向量空间所包含的最大线性无关向量的数目，称作该向量空间的维数。

范数