Монгол Бодлогын Сан

Заавар:

Бодолт: $3\vec{\mathstrut{a}}-2\vec{\mathstrut{b}}=\vec{\mathstrut{u}}$

$2\vec{\mathstrut{a}}+\vec{\mathstrut{b}}=\vec{\mathstrut{v}}$

$\vec{\mathstrut{7a}}= \vec{\mathstrut{u}}+2\vec{\mathstrut{v}}$

$\vec{\mathstrut{a}}= \frac {1}{7}\vec{\mathstrut{u}}+\frac{2}{7}\vec{\mathstrut{v}}$

$\vec{\mathstrut{b}}= \vec{\mathstrut{u}}-2\vec{\mathstrut{a}}=\vec{\mathstrut{u}}-\frac{2}{7}\vec{\mathstrut{u}}-\frac{4}{7}\vec{\mathstrut{v}}=-\frac{2}{7}\vec{\mathstrut{u}}+\frac{3}{7}\vec{\mathstrut{v}}$

$\vec{\mathstrut{a}}\cdot \vec{\mathstrut{b}} = (\vec{\mathstrut{a}}= \frac {1}{7}\vec{\mathstrut{u}}+\frac{2}{7}\vec{\mathstrut{v}})(-\frac{2}{7}\vec{\mathstrut{u}} +\frac{3}{7}\vec{\mathstrut{v}}) =- \frac{2}{49}\vec{\mathstrut{u^2}}- \frac{1}{49}\vec{\mathstrut{uv}}+\frac{6}{49}\vec{\mathstrut{v^2}}-\frac{4}{49}-\frac{1}{49}|\vec{\mathstrut{u}}||\vec{\mathstrut{u}}|\cdot\cos \pi=\frac{4}{49}-\frac{1}{49}\ cos \pi$

$\frac{4}{49}-\frac{1}{49}=\frac{3}{49}<\vec{\mathstrut{a}}\cdot\vec{\mathstrut{b}}<\frac{4}{49}-(-\frac{1}{49})=\frac{5}{49}$