Mathematics - Vector Question with Solution | TestHub

$[(\vec{a}+\vec{b})\times (\vec{b}-\vec{c})(\vec{b}+\vec{c})\times (\vec{c}+\vec{a})(\vec{c}-\vec{a})\times (\vec{a}+\vec{b})]$

$$\begin{array}{rl}& =[\vec{a}\times \vec{b}-\vec{b}\times \vec{c}+\vec{c}\times \vec{a}-\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}-\vec{a}\times \vec{b}-\vec{b}\times \vec{c}+\vec{c}\times \vec{a}]\\ & =[\vec{a}\times \vec{b}\vec{b}\times \vec{c}\vec{c}\times \vec{a}]\left|\begin{array}{ccc}1& -1& 1\\ -1& 1& 1\\ -1& -1& 1\end{array}\right|\\ =& 4[\vec{a}\vec{b}\vec{c}{]}^2\end{array}$$