Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three vectors and $|\overrightarrow{b}|=|\overrightarrow{c}|$ then $(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{c})\}\times (\overrightarrow{b}\times \overrightarrow{c}).(\overrightarrow{b}+\overrightarrow{c})=$

• A. $2$
• B. $-1$
• C. $0$
• D. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

Answer 1

Answer: (C)

$0$

We have

$\left\{\right(\overline{a}+\overline{b})\times (\overline{a}+\overline{c}\left)\right\}\times (\overline{b}\times \overline{c}).(\overline{b}+\overline{c})$
$=\left\{\right(\overline{a}+\overline{b})\times (\overline{a}+\overline{c}\left)\right\}.\left(\overline{c}\right)\overline{b}-\left\{\right(\overline{a}+\overline{b})\times (\overline{a}+\overline{c}\left)\right\}.\left(\overline{b}\right)\overline{c}$
$=\{\overline{a}\times \overline{c}+\overline{b}\times \overline{a}+\overline{b}\times \overline{c}\}.\overline{c}\overline{b}-\{\overline{a}\times \overline{c}+\overline{b}\times \overline{a}+\overline{b}\times \overline{c}\}.\overline{b}\overline{c}$
$=\left\{\right(\overline{b}\times \overline{a}).\overline{c}\}\overline{b}-\left\{\right(\overline{a}\times \overline{c}.\overline{b}\left)\right\}\overline{c}$
$=\left[\overline{a}\overline{c}\overline{b}\right](\overline{b}-\overline{c})$
This vector is dot producted with $(\overline{b}+\overline{c})$.
Hence the answer becomes $0$, since $|\overline{b}|=|\overline{c}|$.