Question

If$\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{0}$ then prove that $k(\overline{a}+\overline{c})=\overline{b},kϵR$

Answer 1

we have,

vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are satisfy conditions

$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}\ne 0$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{c}\times \overrightarrow{b}$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{b}=\overrightarrow{0}$

$\Rightarrow (\overrightarrow{a}+\overrightarrow{c})\times \overrightarrow{b}=\overrightarrow{0}$

Then, $\overrightarrow{a}+\overrightarrow{c}$ are parallel

Now,

$\overrightarrow{b}=k(\overrightarrow{a}+\overrightarrow{c})$

Where k is the scalar.

Hence, this is the answer.