Using the properties of the vector triple product and the scalar triple product,prove that

Question

$$ (\bar{a}\times \bar{b})\cdot \left( (\bar{b}+\bar{c})\times (\bar{c}+\bar{a}) \right) = 0 $$

I tried to solve this but always stuck

Answer 1

I have tried several times several ways but I couldn't approach the solution.this is one way i tried to solve this(https://i.stack.imgur.com/T2hpE.jpg)

Answer 2

This is false for arbitrary vectors $\vec{a},\vec{b},\vec{c}$. Perhaps there is a certain relationship between the vectors you forgot to mention? As a counterexample, take $\vec{a}=(1,0,0), \vec{b}=(0,1,0), \vec{c}=(1,0,0)$.Then $$\vec a\times \vec b=(0,0,1)\\ \vec b+\vec c=(1,1,0)\\ \vec c+\vec a=(2,0,0)\\ (\vec b+\vec c)\times(\vec c+\vec a)=(0,0,-2)\\ (\vec a\times\vec b)\cdot\bigl((\vec b+\vec c)\times(\vec c+\vec a)\bigr)=-2 $$