Question

[Trần Ngọc Hân]

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. \(\overrightarrow{CB}\) - \(\overrightarrow{CA}\)

Answer 1

[Nhã Doanh]

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