Question

6．已知a、b满足|a|=1，|b|=$\sqrt{5}$，|a+b|=|a-b|，则|2a-b|=3．

Answer 1

分析 根据题意得|$\overrightarrow{a}+\overrightarrow{b}$|²=|$\overrightarrow{a}-\overrightarrow{b}$|²，展开即可得到$\overrightarrow{a}•\overrightarrow{b}$的值，再计算$\sqrt{{|\begin{array}{l}{2\overrightarrow{a}-\overrightarrow{b}}\end{array}|}^{2}}$即可．

解答 解：∵|$\overrightarrow{a}+\overrightarrow{b}$|=|$\overrightarrow{a}-\overrightarrow{b}$|，
∴|$\overrightarrow{a}+\overrightarrow{b}$|²=|$\overrightarrow{a}-\overrightarrow{b}$|²，
即${\overrightarrow{a}}^{2}+2\overrightarrow{a}•\overrightarrow{b}+{\overrightarrow{b}}^{2}={\overrightarrow{a}}^{2}-2\overrightarrow{a}•\overrightarrow{b}+{\overrightarrow{b}}^{2}$，
化简得$\overrightarrow{a}•\overrightarrow{b}=0$，
所以$|\begin{array}{l}{2\overrightarrow{a}-\overrightarrow{b}}\end{array}|$=$\sqrt{{|\begin{array}{l}{2\overrightarrow{a}-\overrightarrow{b}}\end{array}|}^{2}}$
=$\sqrt{4{\overrightarrow{a}}^{2}-4\overrightarrow{a}•\overrightarrow{b}+{\overrightarrow{b}}^{2}}$
=$\sqrt{4+5}$
=3，
故答案为：3．

点评 本题考查数量积的运算性质，属基础题．