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