a) G là trọng tâm \(\Delta ABC\Leftrightarrow\left\{{}\begin{matrix}x_G=\dfrac{2+1-1}{3}=\dfrac{2}{3}\\y_G=\dfrac{3+2+2}{3}=\dfrac{7}{3}\\z_G=\dfrac{0+3+0}{3}=1\end{matrix}\right.\)
\(\Rightarrow G\left(\dfrac{2}{3};\dfrac{7}{3};1\right)\Rightarrow\) Đúng
b) \(\overrightarrow{AB}=\left(-1;-1;3\right);\overrightarrow{CD}=\left(3;2;0\right)\)
\(cos\left(\widehat{\overrightarrow{AB};\overrightarrow{CD}}\right)=\dfrac{\left|-1.3+\left(-1\right).2+3.0\right|}{\sqrt{\left(-1\right)^2+\left(-1\right)^2+3^2}.\sqrt{3^2+2^2+0^2}}=\dfrac{5}{\sqrt{11}.\sqrt{13}}=0,42\)
\(\Rightarrow\left(\widehat{\overrightarrow{AB};\overrightarrow{CD}}\right)=65,3^o\Rightarrow\) Sai
c) \(E\left(x;y;z\right)\)
\(2\overrightarrow{ED}+4\overrightarrow{EB}-3\overrightarrow{EC}=\overrightarrow{0}\)
\(\Leftrightarrow2\left(2-x;4-y;0-z\right)+4\left(1-x;2-y;3-z\right)-3\left(-1-x;2-y;0-z\right)=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}2\left(2-x\right)+4\left(1-x\right)-3\left(-1-x\right)=0\\2\left(4-y\right)+4\left(2-y\right)-3\left(2-y\right)=0\\-2z+4\left(3-z\right)+3z=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{11}{3}\\y=\dfrac{10}{3}\\z=4\end{matrix}\right.\) \(\Rightarrow E\left(\dfrac{11}{3};\dfrac{10}{3};4\right)\Rightarrow\) Sai
d) \(\overrightarrow{AH}=\left(x_o-2;y_o-3;z_o\right)\)
\(\overrightarrow{CD}=\left(3;2;0\right)\)
\(\overrightarrow{AH}\perp\overrightarrow{CD}\Leftrightarrow3\left(x_o-2\right)+2\left(y_o-3\right)+0.z\Leftrightarrow3x_o+2y_o=12\left(1\right)\)
\(\overrightarrow{CH}=\left(x_o+1;y_o-2;z_o\right)\)
\(\overrightarrow{AD}=\left(0;1;0\right)\)
\(\overrightarrow{CH}\perp\overrightarrow{AD}\Leftrightarrow y_o-2=0\Leftrightarrow y_o=2\Rightarrow x_o=4-\dfrac{2y_o}{3}=4-\dfrac{2.2}{3}=\dfrac{8}{3}\)
\(\Rightarrow H\left(\dfrac{8}{3};2;z_o\right)\)
\(\overrightarrow{AC}=\left(-3;-1;0\right)\)
Véc tơ pháp tuyến của \(\left(ACD\right):\overrightarrow{n_P}=\left[\overrightarrow{AC};\overrightarrow{AD}\right]=\)\(\left(0;0;3\right)\)
\(\Rightarrow\left(ACD\right):3\left(z-0\right)=0\Leftrightarrow z=0\)
\(H\left(\dfrac{8}{3};2;z_o\right)\in\left(ACD\right)\Rightarrow H\left(\dfrac{8}{3};2;0\right)\)
\(3x_o-y_o+2024z_o=3.\dfrac{8}{3}-2+2024.0=6\)
\(\Rightarrow\) Đúng
