If $4\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}}+9\overrightarrow{\mathrm{c}}=0, \mathrm{then} \left(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right)\times \left[\left(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\right)\times \left(\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}\right)\right]$ is equal to

$\begin{array}{rcl}4\overline{\mathrm{a}}+5\overline{\mathrm{b}}+9\overline{\mathrm{c}}& =& 0\end{array}$
$\begin{array}{rcl} & & \\ & & \end{array}$$\begin{array}{rcl} & & \\ 0+5(\overline{\mathrm{a}}\times \overline{\mathrm{b}})+9(\overline{\mathrm{a}}\times \overline{\mathrm{c}})& =& 0\end{array}$
$\begin{array}{cl}\Rightarrow & 5(\overline{\mathrm{a}}\times \overline{\mathrm{b}})=9 (\overline{\mathrm{c}}\times \overline{\mathrm{a}})\\ & \therefore \overline{\mathrm{a}}\times \overline{\mathrm{b}}, \overline{\mathrm{c}}\times \overline{\mathrm{a}} \mathrm{Parallel}\end{array}$
$\begin{array}{rcl}(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\times \left(\right(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\times (\overline{\mathrm{c}}\times \overline{\mathrm{a}}\left)\right)& =& \overline{0}\end{array}$