Lời giải
a) \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {DE} = \overrightarrow {AE} \)
Cách 1.
Ta có: \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {DE} = \left( {\overrightarrow {AB} + \overrightarrow {BC} } \right) + \overrightarrow {CD} + \overrightarrow {DE} \)
\( = \overrightarrow {AC} + \overrightarrow {CD} + \overrightarrow {DE} = \left( {\overrightarrow {AC} + \overrightarrow {CD} } \right) + \overrightarrow {DE} \)
\( = \overrightarrow {AD} + \overrightarrow {DE} = \overrightarrow {AE} \).
Cách 2.
Ta có: \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {DE} = \left( {\overrightarrow {AB} + \overrightarrow {BC} } \right) + \left( {\overrightarrow {CD} + \overrightarrow {DE} } \right)\)
\( = \overrightarrow {AC} + \overrightarrow {CE} = \overrightarrow {AE} \).
b) \(\overrightarrow {AB} - \overrightarrow {AC} + \overrightarrow {DE} - \overrightarrow {DF} = \overrightarrow {FB} + \overrightarrow {CE} \)
Cách 1.
Ta có: \[\overrightarrow {AB} - \overrightarrow {AC} + \overrightarrow {DE} - \overrightarrow {DF} \]
\[ = \overrightarrow {CB} + \overrightarrow {FE} \]
\[ = \overrightarrow {CB} + \overrightarrow {BE} - \overrightarrow {BE} + \overrightarrow {FE} \]
\[ = \overrightarrow {CB} + \overrightarrow {BE} + \overrightarrow {EB} - \overrightarrow {EF} \]
\[ = \overrightarrow {CE} + \overrightarrow {FB} \].
Cách 2.
Ta có: \[\overrightarrow {AB} - \overrightarrow {AC} + \overrightarrow {DE} - \overrightarrow {DF} - \overrightarrow {FB} - \overrightarrow {CE} \]
\[ = \left( {\overrightarrow {AB} - \overrightarrow {AC} } \right) + \left( {\overrightarrow {DE} - \overrightarrow {DF} } \right) - \overrightarrow {FB} - \overrightarrow {CE} \]
\[ = \overrightarrow {CB} + \overrightarrow {FE} - \overrightarrow {FB} - \overrightarrow {CE} \]
\[ = \left( {\overrightarrow {CB} - \overrightarrow {CE} } \right) + \left( {\overrightarrow {FE} - \overrightarrow {FB} } \right)\]
\[ = \overrightarrow {EB} + \overrightarrow {BE} = \overrightarrow {EE} = \overrightarrow 0 \]
Do đó \[\overrightarrow {AB} - \overrightarrow {AC} + \overrightarrow {DE} - \overrightarrow {DF} - \overrightarrow {FB} - \overrightarrow {CE} = \overrightarrow 0 \]
\( \Leftrightarrow \overrightarrow {AB} - \overrightarrow {AC} + \overrightarrow {DE} - \overrightarrow {DF} = \overrightarrow {FB} + \overrightarrow {CE} \).
