微积分学习笔记四：空间向量基础

1、内积和外积：设$\vec{a}=(x_{a},y_{a},z_{a}),\vec{b}=(x_{b},y_{b},z_{b})$

（1）内积：$\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|cos \varphi =\sqrt{x_{a}^{^{2}}+y_{a}^{^{2}}+z_{a}^{^{2}}}\sqrt{x_{b}^{^{2}}+y_{b}^{^{2}}+z_{b}^{^{2}}}cos \varphi =x_{a}x_{b}+y_{a}y_{b}+z_{a}z_{b}?$

（2）外积：$\vec{a}\times \vec{b}=(y_{a}z_{b}-z_{a}y_{b},z_{a}x_{b}-x_{a}z_{b},x_{a}y_{b}-y_{a}x_{b})$

2、平面的表示方式：$Ax+By+Cx+D=0$，法向$\vec{n}=(A,B,C)$

3、两平面的夹角：

$\Gamma _{1}:A_{1}x+B_{1}y+C_{1}z+D_{1}=0$

$\Gamma _{2}:A_{2}x+B_{2}y+C_{2}z+D_{2}=0$

$cos \theta=\frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt{A_{1}^{_{2}}+B_{1}^{_{2}}+C_{1}^{_{2}}}\sqrt{A_{2}^{_{2}}+B_{2}^{_{2}}+C_{2}^{_{2}}}}$

4、点到面的距离:面$Ax+By+Cz+D=0$,点$P(x_{1},y_{1},z_{1})$

\[d=\frac{|Ax_{1}+By_{1}+Cz_{1}+D|}{\sqrt {A^{^{2}}+B^{^{2}}+C^{^{2}}}}\]

5、空间直线：方向$\vec{n}=(l,m,n)$,经过点$P=(x_{0},y_{0},z_{0})$

标准方程：$\frac{x-x_{0}}{l}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n}$

参数方程：$\left\{\begin{matrix}\\x=x_{0}+lt\\y=y_{0}+mt\\z=z_{0}+nt\end{matrix}\right.$

5、空间两直线的夹角：

$L_{1}:\frac{x-x_{1}}{l_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}}$

$L_{2}:\frac{x-x_{2}}{l_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}$

$cos \varphi =\frac{\vec v_{1}\cdot \vec v_{2}}{|\vec v_{1}||\vec v_{2}|}=\frac{l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}}{\sqrt{l_{1}^{^2}+m_{1}^{^2}+n_{1}^{^2}}\sqrt{l_{2}^{^2}+m_{2}^{^2}+n_{2}^{^2}}}$

6、空间直线与平面的夹角：

$\Gamma :Ax+By+Cz+D=0$

$L:\frac{x-x_{0}}{l}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n}$

$sin \varphi=\frac{|Al+Bm+Cn|}{\sqrt{A^{^{2}}+C^{^{2}}+B^{^{2}}}\sqrt{l^{^{2}}+m^{^{2}}+n^{^{2}}}}$