Cho tam giác ABC. Gọi I là điểm thoả mãn \(\overrightarrow{BI}=k\overrightarrow{BC}\left(k\ne1\right)\). Biểu diễn \(\overrightarrow{AI}\) qua hai vectơ \(\overrightarrow{AB},\overrightarrow{AC}\).
\(\overrightarrow{AI}=\left(k-1\right)\overrightarrow{AB}-k\overrightarrow{AC}\).\(\overrightarrow{AI}=\left(1-k\right)\overrightarrow{AB}+k\overrightarrow{AC}\).\(\overrightarrow{AI}=\left(1+k\right)\overrightarrow{AB}-k\overrightarrow{AC}\).\(\overrightarrow{AI}=\left(1+k\right)\overrightarrow{AB}+k\overrightarrow{AC}\).Hướng dẫn giải:\(\overrightarrow{AI}=\overrightarrow{AB}+\overrightarrow{BI}\)
\(\overrightarrow{BC}=\overrightarrow{BA}+\overrightarrow{AC}=\dfrac{1}{k}\overrightarrow{BI}\)
\(\Rightarrow\overrightarrow{BI}=k\overrightarrow{BA}+k\overrightarrow{AC}\)
\(\Rightarrow\overrightarrow{AI}=\overrightarrow{AB}+k\overrightarrow{BA}+k\overrightarrow{AC}=\left(1-k\right)\overrightarrow{AB}+k\overrightarrow{AC}\)
Vậy \(\overrightarrow{AI}=\left(1-k\right)\overrightarrow{AB}+k.\overrightarrow{AC}\)
