If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three non-zero vectors satisfying the condition $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$ and $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ then which of the following always hold(s) good?

Clearly $\overrightarrow{a}.\overrightarrow{c}=0$ and $\overrightarrow{b}.\overrightarrow{c}=0$ 

Also $\overrightarrow{a}.\overrightarrow{b}=0\Rightarrow \left(a\right)$

$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$

Dot with $\overrightarrow{b}\Rightarrow \overrightarrow{b}.\overrightarrow{c}=0;$ similarly $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$

Dot with $\overrightarrow{a}=\widehat{i};\overrightarrow{b}=\widehat{j}and\overrightarrow{c}=\widehat{k}$

Dot with $\overrightarrow{c}\Rightarrow \overrightarrow{a}.\overrightarrow{c}=0\Rightarrow \overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$

Again $\frac{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}{\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|}=\frac{\left|\overrightarrow{c}\right|}{\left|\overrightarrow{a}\right|}\Rightarrow \frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{c}\right|}=\frac{\left|\overrightarrow{c}\right|}{\left|\overrightarrow{a}\right|}\Rightarrow \left|\overrightarrow{a}\right|=\left|\overrightarrow{c}\right|and\left|\overrightarrow{b}\right|=1\Rightarrow \overrightarrow{a}\times \overrightarrow{b}.\overrightarrow{c}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|={\left|\overrightarrow{a}\right|}^2={\left|\overrightarrow{c}\right|}^2$

(Children will assume $\overrightarrow{a}=\widehat{i};\overrightarrow{b}=\widehat{j}and\overrightarrow{c}=\widehat{k}$ but in this case all the four will be correct which will be wrong)