Question

$\overline{a}\times \left\{\overline{a}\times \left(\overline{a}\times \overline{b}\right)\right\}$ is equal to

Answer 1

$\begin{array}{c}\overrightarrow{a}\times \left\{\overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)\right\}=\\ =\overrightarrow{a}\times \left\{\left(\overrightarrow{a}\cdot \overrightarrow{b}\right)\overrightarrow{a}-\left(\overrightarrow{a}\cdot \overrightarrow{a}\right)\overrightarrow{b}\right\}\\ =\left(\overrightarrow{a}\cdot \overrightarrow{b}\right)\left(\overrightarrow{a}\times \overrightarrow{a}\right)-\left(\overrightarrow{a}\cdot \overrightarrow{a}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)\\ =\left(\overrightarrow{a}\cdot \overrightarrow{a}\right)\left(\overrightarrow{b}\times \overrightarrow{a}\right)\end{array}$