Cho 5 điểm A, B, C, D, E. Chứng minh rằng:

a) $\overrightarrow {AB} + \overrightarrow {CD} + \overrightarrow {E{\rm{A}}} = \overrightarrow {CB} + \overrightarrow {E{\rm{D}}}$.

b) $\overrightarrow {AC} + \overrightarrow {CD} - \overrightarrow {EC} = \overrightarrow {A{\rm{E}}} - \overrightarrow {DB} + \overrightarrow {CB}$.

a) Ta có: $\overrightarrow {AB} + \overrightarrow {CD} + \overrightarrow {E{\rm{A}}} = \overrightarrow {CB} + \overrightarrow {E{\rm{D}}}$

$\Leftrightarrow \overrightarrow {AB} + \overrightarrow {CD} + \overrightarrow {E{\rm{A}}} + \overrightarrow {BC} + \overrightarrow {{\rm{DE}}} = \overrightarrow 0$

$\Leftrightarrow (\overrightarrow {AB} + \overrightarrow {BC} ) + (\overrightarrow {CD} + \overrightarrow {{\rm{DE}}} ) + \overrightarrow {E{\rm{A}}} = \overrightarrow 0$

$\Leftrightarrow \overrightarrow {AC} + \overrightarrow {CE} + \overrightarrow {E{\rm{A}}} = \overrightarrow 0$

$\Leftrightarrow \overrightarrow {AE} + \overrightarrow {E{\rm{A}}} = \overrightarrow 0$ (luôn đúng)

Suy ra $\overrightarrow {AB} + \overrightarrow {CD} + \overrightarrow {E{\rm{A}}} = \overrightarrow {CB} + \overrightarrow {E{\rm{D}}}$

b) Ta có: $\overrightarrow {AC} + \overrightarrow {CD} - \overrightarrow {EC} = (\overrightarrow {AC} + \overrightarrow {CE} ) + \overrightarrow {CD} = \overrightarrow {AE} + \overrightarrow {C{\rm{D}}}$

$= \overrightarrow {AE} + (\overrightarrow {CB} + \overrightarrow {B{\rm{D}}} ) = \overrightarrow {AE} - \overrightarrow {DB} + \overrightarrow {CB}$

Vậy $\overrightarrow {AC} + \overrightarrow {CD} - \overrightarrow {EC} = \overrightarrow {A{\rm{E}}} - \overrightarrow {DB} + \overrightarrow {CB}$.