$\mathrm{lf} \overline{\mathrm{a}}+2 \overline{\mathrm{b}}+4 \overline{\mathrm{c}}= \overline{0}, \mathrm{then} \overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{b}}\times \overline{\mathrm{c}}+ \overline{\mathrm{c}}\times \overline{\mathrm{a}}=$

$\overline{\mathrm{a}}+2\overline{\mathrm{b}}+4\overline{\mathrm{c}}=\overline{0}$

Cross Product with $\overline{\mathrm{a}}$

$\begin{array}{rcl}\overline{\mathrm{a}}\times \overline{\mathrm{a}}+\overline{\mathrm{a}}\times 2\overline{\mathrm{b}}+\overline{\mathrm{a}}\times 4\overline{\mathrm{c}}& =& 0\\ 0+2(\overline{\mathrm{a}}\times \overline{\mathrm{b}})+4(\overline{\mathrm{a}}\times \overline{\mathrm{c}})& =& 0\\ & \Rightarrow & \overline{\mathrm{a}}\times \overline{\mathrm{b}}=2(\overline{\mathrm{c}}\times \overline{\mathrm{a}})\end{array}$

Cross Product with $\overline{\mathrm{b}}$

$\begin{array}{rcl}\overline{\mathrm{b}}\times \overline{\mathrm{a}}+0+4\overline{\mathrm{b}}\times \overline{\mathrm{c}}& =& 0\\ \overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& 4(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\\ \overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{b}}\times \overline{\mathrm{c}}+\overline{\mathrm{c}}\times \overline{\mathrm{a}}& =& 4(\overline{\mathrm{b}}\times \overline{\mathrm{c}})+\overline{\mathrm{b}}\times \overline{\mathrm{c}}+2(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\\ & =& 7(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\end{array}$