Question

Compute $\overline{a}\times (\overline{b}+\overline{c})+\overline{b}\times (\overline{c}+\overline{a})+\overline{c}\times (\overline{a}+\overline{b})$.

Answer 1

From distribuitive law,

$\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}$

Similarly, $\overrightarrow{b}\times (\overrightarrow{c}+\overrightarrow{a})=\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{a}$

$\overrightarrow{c}\times (\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{c}\times \overrightarrow{a}$

Adding them, $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=0$

Hence, this is the answer.