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Câu 1:
\(\overrightarrow{BI}=\frac{1}{2}\overrightarrow{BD}+\frac{1}{2}\overrightarrow{BC}\\ =\frac{1}{2}\left(\overrightarrow{BA}+\overrightarrow{AD}\right)+\frac{1}{2}\overrightarrow{BC}\\ =\frac{1}{2}\left(-\overrightarrow{AB}+\overrightarrow{AD}\right)+\frac{1}{2}\overrightarrow{AD}\\ =-\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}+\frac{1}{2}\overrightarrow{AD}\\ =-\frac{1}{2}\overrightarrow{AB}+\overrightarrow{AD}=-\frac{1}{2}\overrightarrow{a}+\overrightarrow{b}\)
\(\overrightarrow{CG}=\frac{1}{3}\overrightarrow{CC}+\frac{1}{3}\overrightarrow{CB}+\frac{1}{3}\overrightarrow{CD}\\ =-\frac{1}{3}\overrightarrow{AD}-\frac{1}{3}\overrightarrow{AB}=-\frac{1}{3}\overrightarrow{b}-\frac{1}{3}\overrightarrow{a}\)
\(a\text{) }\overrightarrow{DE}=\overrightarrow{DA}+\overrightarrow{AE}=-2\overrightarrow{AB}+\frac{2}{5}\overrightarrow{AC}\\ \overrightarrow{DG}=\overrightarrow{DA}+\overrightarrow{AG}\\ =-2\overrightarrow{AB}+\frac{1}{3}\left(\overrightarrow{AA}+\overrightarrow{AB}+\overrightarrow{AC}\right)\\ =-2\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}\\ =-\frac{5}{3}\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}\)
\(\text{b) }\overrightarrow{DG}=-\frac{5}{3}\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}=\frac{5}{6}\left(-2\overrightarrow{AB}+\frac{2}{5}\overrightarrow{AC}\right)=\frac{5}{6}\overrightarrow{DE}\)
=> D;G;E thẳng hàng
c) \(\overrightarrow{KA}+\overrightarrow{KB}+3\overrightarrow{KC}=2\overrightarrow{KD}\)
\(\Rightarrow\overrightarrow{KA}+\overrightarrow{KB}+\overrightarrow{KC}=2\overrightarrow{KD}-2\overrightarrow{KC}\\ \Rightarrow3\overrightarrow{KG}=2\left(\overrightarrow{KD}-\overrightarrow{KC}\right)\\ \Rightarrow3\overrightarrow{KG}=2\overrightarrow{CD}\\ \Rightarrow\overrightarrow{KG}=\frac{2}{3}\overrightarrow{CD}\\ \Rightarrow\overrightarrow{KG}\text{ cùng phương }\overrightarrow{CD}\\ \Rightarrow KG//CD\)
\(X=\left\{e,c,d\right\}\\ X=\left\{g,c,d\right\}\\ X=\left\{e,g,c,d\right\}\)
`X \\ B = C <=> X = B ∪ C`
`B ∪ C = {a,b,c,d,e,g}`
`=> X={a,b,c,d,e,g}`.