# 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：

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

![](https://i.postimg.cc/qBDr87Y2/image.png)

## 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