Question

5．已知向量$\overrightarrow{a}$，$\overrightarrow{b}$，满足$\overrightarrow{a}$•$\overrightarrow{b}$=0，|$\overrightarrow{a}$|=2，|$\overrightarrow{b}$|=1，则|$\overrightarrow{a}$+2$\overrightarrow{b}$|=4$\sqrt{2}$．

Answer 1

分析 根据题意，由数量积的运算性质可得|$\overrightarrow{a}$+2$\overrightarrow{b}$|²=（$\overrightarrow{a}$+2$\overrightarrow{b}$）²=$\overrightarrow{a}$²+4$\overrightarrow{a}$•$\overrightarrow{b}$+4$\overrightarrow{b}$²=|$\overrightarrow{a}$|²+4$\overrightarrow{a}$•$\overrightarrow{b}$+4|$\overrightarrow{b}$|²，代入数据可得|$\overrightarrow{a}$+2$\overrightarrow{b}$|²的值，进而可得答案．

解答 解：根据题意，|$\overrightarrow{a}$+2$\overrightarrow{b}$|²=（$\overrightarrow{a}$+2$\overrightarrow{b}$）²=$\overrightarrow{a}$²+4$\overrightarrow{a}$•$\overrightarrow{b}$+4$\overrightarrow{b}$²=|$\overrightarrow{a}$|²+4$\overrightarrow{a}$•$\overrightarrow{b}$+4|$\overrightarrow{b}$|²=8，
则|$\overrightarrow{a}$+2$\overrightarrow{b}$|=4$\sqrt{2}$，
故答案为：4$\sqrt{2}$．

点评 本题考查平面向量数量积的运算，掌握数量积的有关运算性质是解题的关键．