EFGH
A,B,C,D,E,F,G
H
(A, \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})
\overrightarrow{AG}, \overrightarrow{CF}\overrightarrow{CH}
( \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})
(AG)
(CFH)
K
(AG)
(CFH)
A
(CFH)
A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0), E(0,0,1), F(1,0,1), G(1,1,1), H(0,1,1)
\overrightarrow{AG}(1,1,1), \overrightarrow{CF}(0,-1,1), \overrightarrow{CH}(-1,0,1)
\overrightarrow{AG}.\overrightarrow{CF} = 1\times 0+1\times-1+1\times 1 = -1+1 =0
\overrightarrow{AG}.\overrightarrow{CH} = 1\times -1+1\times0+1\times 1 = -1+1 =0
\begin{cases} \overrightarrow{AG}.\overrightarrow{CH} = 0 \\ \overrightarrow{AG}.\overrightarrow{CF} = 0\end{cases} \Rightarrow \begin{cases} (AG)\perp (CF) \\ (AG) \perp (CH) \end{cases} \Rightarrow (AG) \perp (CFH)
(CF)(CH)
(CFH)
(AG)(CFH)
(CFH)
\overrightarrow{AG}(ACF): x+y+z+d=0C(1,1,0)\in (CFH) \Longleftrightarrow 1+1+0+d=0 \Longleftrightarrow d=-2
(CFH): x+y+z-2=0
(AG): \begin{cases} x=t \\ y=t \\ z=t \end{cases} t\in \mathbb{R}(AG)\cap (CFH): t+t+t-2=0 \Longleftrightarrow t=\dfrac{2}{3}
K(\dfrac{2}{3},\dfrac{2}{3},\dfrac{2}{3})
A(CFH)AK = \sqrt{\dfrac{4}{9}+\dfrac{4}{9}+\dfrac{4}{9}} = \sqrt{\dfrac{12}{9}} = \dfrac{2\sqrt{3}}{3}
ABCD
