Question

cho 2 vecto đơn vị thỏa mãn |\(\overrightarrow{a}+\overrightarrow{b}\)|=2 tính ( \(\overrightarrow{3a}-\overrightarrow{4b}\))(\(\overrightarrow{2a}+\overrightarrow{5b}\))

Answer 1

Lời giải:
Vì $\overrightarrow{a}, \overrightarrow{b}$ là 2 vecto đơn vị nên:

\(|\overrightarrow{a}|=|\overrightarrow{b}|=1\)

Ta có:

\(|\overrightarrow{a}+\overrightarrow{b}|=2\Rightarrow (\overrightarrow{a}+\overrightarrow{b})^2=4\)

\(\Leftrightarrow |\overrightarrow{a}|^2+2\overrightarrow{a}.\overrightarrow{b}+|\overrightarrow{b}|^2=4\Rightarrow \overrightarrow{a}.\overrightarrow{b}=1\)

Do đó:

\((3\overrightarrow{a}-4\overrightarrow{b})(2\overrightarrow{a}+5\overrightarrow{b})=6(\overrightarrow{a})^2+7\overrightarrow{a}.\overrightarrow{b}-20(\overrightarrow{b})^2\)

\(=6|\overrightarrow{a}|^2+7\overrightarrow{a}.\overrightarrow{b}-20|\overrightarrow{b}|^2=6+7.1-20=-7\)