Question

Show that vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if $\overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c},\overrightarrow{c}+\overrightarrow{a}$ are coplanar

Answer 1

$[\overline{a}+\overline{b}\overline{b}+\overline{c}\overline{c}+\overline{a}]=0$

$[\overline{a}-\overline{c}\overline{b}+\overline{c}\overline{c}+\overline{a}]=0$

$[2\overline{a}\overline{b}+\overline{c}\overline{c}+\overline{a}]=0$

$[\overline{a}\overline{b}+\overline{c}\overline{c}+\overline{a}]=0$

$[\overline{a}\overline{b}-\overline{c}\overline{c}+\overline{a}]=0$

$[\overline{a}\overline{b}\overline{c}+\overline{a}]=0$

$\left[\overline{a}\overline{b}\overline{c}\right]+\left[\overline{a}\overline{b}\overline{}a\right]=0$

$\left[\overline{a}\overline{b}\overline{c}\right]=0$

$\therefore \overline{a},\overline{b}$ & $\overline{c}$ are coplnar