\(\overrightarrow{AB}=\left(1-a;b;0\right);\overrightarrow{AC}=\left(1-a;0;c\right);\overrightarrow{HC}=\left(-2;-2;c-1\right);\overrightarrow{HB}=\left(-2;b-2;-1\right)\)
\(\left\{{}\begin{matrix}\overrightarrow{AB}.\overrightarrow{HC}=0\\\overrightarrow{AC}.\overrightarrow{HB}=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}-2\left(1-a\right)-2b=0\\-2\left(1-a\right)-c=0\end{matrix}\right.\) \(\Rightarrow c=2b=2\left(a-1\right)\)
\(\left[\overrightarrow{AB};\overrightarrow{AC}\right]=\left(bc;c\left(a-1\right);b\left(a-1\right)\right)=\left(2\left(a-1\right)^2;2\left(a-1\right)^2;\left(a-1\right)^2\right)=\left(a-1\right)^2.\left(2;2;1\right)\)
A;B;C;H đồng phẳng
\(\Rightarrow\left[\overrightarrow{AB};\overrightarrow{AC}\right].\overrightarrow{HB}=0\Rightarrow2.\left(-2\right)+2.\left(-2\right)+1.\left(c-1\right)=0\)
\(\Rightarrow c=9\Rightarrow b=\dfrac{9}{2}\Rightarrow a=\dfrac{11}{2}\)