Question

4．正四面體ABCD邊長為a，點(diǎn)E、F分別是BC、AD的中點(diǎn)，則$\begin{array}{l}→\\{AE}\end{array}•\begin{array}{l}→\\{AF}\end{array}$的值為（　　）

• A． a²
• B． $\frac{1}{2}{a^2}$
• C． $\frac{1}{4}{a^2}$
• D． $\frac{{\sqrt{3}}}{4}{a^2}$

Answer 1

分析 如圖所示，$\overrightarrow{AF}=\frac{1}{2}\overrightarrow{AD}$，$\overrightarrow{AE}=\frac{1}{2}（\overrightarrow{AB}+\overrightarrow{AC}）$，代入$\begin{array}{l}→\\{AE}\end{array}•\begin{array}{l}→\\{AF}\end{array}$，利用數(shù)量積運(yùn)算性質(zhì)即可得出．

解答 解：如圖所示，
$\overrightarrow{AF}=\frac{1}{2}\overrightarrow{AD}$，$\overrightarrow{AE}=\frac{1}{2}（\overrightarrow{AB}+\overrightarrow{AC}）$，
∴$\begin{array}{l}→\\{AE}\end{array}•\begin{array}{l}→\\{AF}\end{array}$=$\frac{1}{4}$$（\overrightarrow{AD}•\overrightarrow{AB}+\overrightarrow{AD}•\overrightarrow{AC}）$
=$\frac{1}{4}$（a²cos60°+a²cos60°）
=$\frac{1}{4}{a}^{2}$．
故選：C．

點(diǎn)評 本題考查了數(shù)量積運(yùn)算性質(zhì)、向量平行四邊形法則，考查了推理能力與計(jì)算能力，屬于中檔題．