$\begin{array}{l}
a)\overrightarrow {AB} + \overrightarrow {DC} + \overrightarrow {BD} + \overrightarrow {CA} \\
= \left( {\overrightarrow {AB} + \overrightarrow {BD} } \right) + \left( {\overrightarrow {DC} + \overrightarrow {CA} } \right)\\
= \overrightarrow {AD} + \overrightarrow {DA} = \overrightarrow 0 \\
\overrightarrow {AB} + \overrightarrow {CD} + \overrightarrow {BC} + \overrightarrow {DA} \\
= \left( {\overrightarrow {AB} + \overrightarrow {BC} } \right) + \left( {\overrightarrow {CD} + \overrightarrow {DA} } \right)\\
= \overrightarrow {AC} + \overrightarrow {CA} = \overrightarrow 0 \\
b)\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} \\
= \overrightarrow {AE} + \overrightarrow {ED} + \overrightarrow {BF} + \overrightarrow {FE} + \overrightarrow {CD} + \overrightarrow {DF} \\
= \left( {\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} } \right) + \left( {\overrightarrow {ED} + \overrightarrow {DF} + \overrightarrow {FE} } \right)\\
= \left( {\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} } \right) + \left( {\overrightarrow {EF} + \overrightarrow {FE} } \right)\\
= \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD}
\end{array}$
