Question

The vectors $\overline{a},\overline{b},\overline{c}$ are coplanar, if and only if $\overline{a}+\overline{b},\overline{b}+\overline{c}$ and $\overline{c}+\overline{a}$ are coplaner

• A. True
• B. False

Answer 1

The correct option is A True

Three vectors $\overline{x},\overline{y}and\overline{z}$ are coplanar.

$If:\overline{x}(\overline{y}\times \overline{z})=0\left[\begin{array}{ccc}x& y& z\end{array}\right]=0$

Here, $(\overline{a}+\overline{b})\left[\right(\overline{b}+\overline{c})\times (\overline{c}+\overline{a}\left)\right]=0$

$⟹(\overline{a}+\overline{b}).\left[\right(\overline{b}\times \overline{c})+(\overline{b}\times \overline{a})+(\overline{c}\times \overline{c})+(\overline{c}\times \overline{a}\left)\right]=0$

$⟹(\overline{a}+\overline{b}).\left[\right(\overline{b}\times \overline{c})+(\overline{b}\times \overline{a})+0+(\overline{c}\times \overline{a}\left)\right]$

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

$+\overline{b}.(\overline{b}\times \overline{a})+\overline{b}.(\overline{c}\times \overline{a})=0$

$⟹\overline{a}.(\overline{b}\times \overline{c})+0+0+0+0+\overline{b}.(\overline{c}\times \overline{a})=0-\left(i\right)$

$\because \left[\begin{array}{ccc}\overline{a}& \overline{b}& \overline{c}\end{array}\right]=\left[\begin{array}{ccc}\overline{b}& \overline{a}& \overline{c}\end{array}\right]$

$⟹\overline{a}.(\overline{b}\times \overline{c})=\overline{b}.(\overline{a}\times \overline{c})$

$⟹\overline{a}.(\overline{b}\times \overline{c})=-\overline{b}.(\overline{c}\times \overline{a})$

Putting in $\left(i\right):$

$\overline{a}.(\overline{b}\times \overline{c})-\overline{a}.(\overline{b}\times \overline{c})=0$

$\therefore If\overline{a}+\overline{b},\overline{b}+\overline{c}and\overline{c}+\overline{a}$ are coplanar, hence $\overline{a},\overline{b}and\overline{c}$ are coplanar too.

$\therefore A)True.$