a: \(\overrightarrow{MA}=\frac12\cdot\overrightarrow{MB}\)
=>A là trung điểm của MB
=>\(\overrightarrow{MA}=\overrightarrow{AB}\)
\(\overrightarrow{CA}+\overrightarrow{MA}=\overrightarrow{CA}+\overrightarrow{AB}=\overrightarrow{CB}\)
b: \(\overrightarrow{NC}=-3\cdot\overrightarrow{NA}\)
=>NC=3NA và N nằm giữa C và A
AC=AN+NC
=AN+3AN
=4AN
=>\(\overrightarrow{AN}=\frac14\cdot\overrightarrow{AC};\overrightarrow{NC}=\frac34\cdot\overrightarrow{AC}\)
3KB=2KC
=>\(KB=\frac23KC\)
KB+KC=BC
=>\(BC=KC+\frac23KC=\frac53KC\)
=>\(BK=\frac25\cdot BC;CK=\frac35\cdot BC\)
\(\overrightarrow{MN}=\overrightarrow{MA}+\overrightarrow{AN}=\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}\)
\(\overrightarrow{MK}=\overrightarrow{MB}+\overrightarrow{BK}=2\cdot\overrightarrow{AB}+\frac25\cdot\overrightarrow{BC}\)
\(=2\cdot\overrightarrow{AB}+\frac25\left(\overrightarrow{BA}+\overrightarrow{AC}\right)=\frac85\cdot\overrightarrow{AB}+\frac25\cdot\overrightarrow{AC}=\frac25\left(4\cdot\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(=\frac85\cdot\left(\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}\right)\)
=>\(\overrightarrow{MK}=\frac85\cdot\overrightarrow{MN}\)
=>M,K,N thẳng hàng
