向量运算
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。