Question

For any three vectors $\overrightarrow{a,}\overrightarrow{b}and\overrightarrow{c}$write the value of $\overrightarrow{a}\times \left(\overrightarrow{b}+\overrightarrow{c}\right)+\overrightarrow{b}\times \left(\overrightarrow{c}+\overrightarrow{a}\right)+\overrightarrow{c}\times \left(\overrightarrow{a}+\overrightarrow{b}\right).$

Answer 1

$$\begin{gathered} \overrightarrow{a}\times \left(\overrightarrow{b}+\overrightarrow{c}\right)+\overrightarrow{b}\times \left(\overrightarrow{c}+\overrightarrow{a}\right)+\overrightarrow{c}\times \left(\overrightarrow{a}+\overrightarrow{b}\right) \\ =\left(\overrightarrow{a}\times \overrightarrow{b}\right)+\left(\overrightarrow{a}\times \overrightarrow{c}\right)+\left(\overrightarrow{b}\times \overrightarrow{c}\right)+\left(\overrightarrow{b}\times \overrightarrow{a}\right)+\left(\overrightarrow{c}\times \overrightarrow{a}\right)+\left(\overrightarrow{c}\times \overrightarrow{b}\right) \\ =\left(\overrightarrow{a}\times \overrightarrow{b}\right)+\left(\overrightarrow{a}\times \overrightarrow{c}\right)+\left(\overrightarrow{b}\times \overrightarrow{c}\right)-\left(\overrightarrow{a}\times \overrightarrow{b}\right)-\left(\overrightarrow{a}\times \overrightarrow{c}\right)-\left(\overrightarrow{b}\times \overrightarrow{c}\right) \\ =\overrightarrow{0} \end{gathered}$$