Question

For any two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b},\left(2\overrightarrow{a}+3\overrightarrow{b}\right)\times \left(5\overrightarrow{a}+7\overleftrightarrow{b}\right)$ = _________________.

Answer 1

$$\begin{gathered} \mathrm{For}\mathrm{any}\mathrm{two}\mathrm{vectors},\overrightarrow{a}\mathrm{and}\overrightarrow{b} \\ \left(2\overrightarrow{a}+3\overrightarrow{b}\right)\times \left(5\overrightarrow{a}+7\overrightarrow{b}\right) \\ =2\overrightarrow{a}\times 5\overrightarrow{a}+3\overrightarrow{b}\times 5\overrightarrow{a}+2\overrightarrow{a}\times 7\overrightarrow{b}+3\overrightarrow{b}\times 7\overrightarrow{b} \\ =10\overrightarrow{a}\times \overrightarrow{a}+15\overrightarrow{b}\times \overrightarrow{a}+14\overrightarrow{a}\times \overrightarrow{b}+21\overrightarrow{b}\times \overrightarrow{b} \\ \mathrm{Since}\overrightarrow{a}\times \overrightarrow{a}=0=\overrightarrow{b}\times \overrightarrow{b}\mathrm{and}\overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{b}\times \overrightarrow{a} \\ =0+15\overrightarrow{b}\times \overrightarrow{a}-14\overrightarrow{b}\times \overrightarrow{a}+0 \\ \left(2\overrightarrow{a}+3\overrightarrow{b}\right)\times \left(5\overrightarrow{a}+7\overrightarrow{b}\right)=\overrightarrow{b}\times \overrightarrow{a} \end{gathered}$$