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：$vec{a} imes vec{b}=(y_{a}z_{b}-z_{a}y_{b},z_{a}x_{b}-x_{a}z_{b},x_{a}y_{b}-y_{a}x_{b})$
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：
$Gamma _{1}:A_{1}x+B_{1}y+C_{1}z+D_{1}=0$
$Gamma _{2}:A_{2}x+B_{2}y+C_{2}z+D_{2}=0$
$cos heta=frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{sqrt{A_{1}^{_{2}}+B_{1}^{_{2}}+C_{1}^{_{2}}}sqrt{A_{2}^{_{2}}+B_{2}^{_{2}}+C_{2}^{_{2}}}}$
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：$left{egin{matrix}\x=x_{0}+lt\y=y_{0}+mt\z=z_{0}+ntend{matrix}
ight.$
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：
$Gamma :Ax+By+Cz+D=0$
$L:frac{x-x_{0}}{l}=frac{y-y_{0}}{m}=frac{z-z_{0}}{n}$
$sin varphi=frac{|Al+Bm+Cn|}{sqrt{A^{^{2}}+C^{^{2}}+B^{^{2}}}sqrt{l^{^{2}}+m^{^{2}}+n^{^{2}}}}$