Cho tam giác ABC và I thỏa mãn \(\overrightarrow{IA}\) = 3. \(\overrightarrow{IB}\) . Đẳng thức nào sau đây đúng ?
A. \(\overrightarrow{CI}\) = \(\overrightarrow{CA}\) - 3. \(\overrightarrow{CB}\)
B. \(\overrightarrow{CI}\) = \(\dfrac{1}{2}\) ( 3. \(\overrightarrow{CB}\) - \(\overrightarrow{CA}\) )
C. \(\overrightarrow{CI}\) = \(\dfrac{1}{2}\) ( \(\overrightarrow{CA}\) - 3. \(\overrightarrow{CB}\) )
D. \(\overrightarrow{CI}\) = 3. \(\overrightarrow{CB}\) - \(\overrightarrow{CA}\)
Ta có: \(\overrightarrow{IA}=3.\overrightarrow{IB}\)
\(\overrightarrow{AB}=2.\overrightarrow{BI}\)
\(\Rightarrow\overrightarrow{BI}=\dfrac{1}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{CB}+\overrightarrow{BI}=\overrightarrow{CB}+\dfrac{1}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{CI}=\overrightarrow{CA}+\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{CI}=\overrightarrow{CA}+\dfrac{3}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{CI}=\overrightarrow{CA}+\dfrac{3}{2}\overrightarrow{AC}+\dfrac{3}{2}\overrightarrow{CB}\)
\(\Rightarrow\overrightarrow{CI}=\overrightarrow{CA}-\dfrac{3}{2}\overrightarrow{CA}+\dfrac{3}{2}\overrightarrow{CB}\)
\(\Rightarrow\overrightarrow{CI}=\dfrac{-1}{2}\overrightarrow{CA}+\dfrac{3}{2}\overrightarrow{CB}\)
\(\Rightarrow\overrightarrow{CI}=\dfrac{1}{2}\left(3\overrightarrow{CB}-\overrightarrow{CA}\right)\)
\(\Rightarrow\) Đáp án B đúng