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:

uv=uxvx+uyvy+uzvz=uvcos(u,v)\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:

w=u×v=[ijkuxuyuzvxvyvz]\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:

u=uxi+uyj+uzk,v=vxi+vyj+vzk\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:

u×v=v×uu×(v×w)=(uw)v(uv)w\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

[uvw]=(u×v)w=u(v×w)=uxuyuzvxvyvzwxwywz=uxvxwxuyvywyuzvzwz\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:

xy=[x1y1x1y2x1ymx2y1x2y2x2ymxny1xny2xnym]\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