Question

If $\overrightarrow{b}$ and $\overrightarrow{c}$ are two non-collinear vectors such that $\overrightarrow{a}$ || $(\overrightarrow{b}\times \overrightarrow{c})$, then $(\overrightarrow{a}\times \overrightarrow{b})$. $(\overrightarrow{a}\times \overrightarrow{c})$ is equal to

• A. $\overrightarrow{a^2}(\overrightarrow{b}.\overrightarrow{c})$
• B. $\overrightarrow{b^2}(\overrightarrow{a}.\overrightarrow{c})$
• C. $\overrightarrow{c^2}(\overrightarrow{a}.\overrightarrow{b})$
• D. $Noneoftheabove$

Answer 1

The correct option is D $Noneoftheabove$

$\overrightarrow{a}\times (\overrightarrow{b}.\overrightarrow{c})$

$\overrightarrow{a}=\lambda (\overrightarrow{b}\times \overrightarrow{c})$

$\lambda (\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{b}.\lambda (\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{c}$

$\Rightarrow \left(\lambda \right(b.b)\overline{c}-\lambda (\overline{b}.\overline{c}\left)\overline{b}\right).\left(\lambda \right(\overline{b}.\overline{c})\overline{c}-(\overline{c}.\overline{c}\left)\overline{b}\right)$

${\lambda }^2|c{|}^2|b{|}^2(\overline{b}.\overline{c}-|c{|}^2|b{|}^2)(b.c)-(\overline{b}.\overline{c}{)}^3+|c{|}^2(\overline{b}.\overline{c})|b{|}^2$

$={\lambda }^2(b.c)|c{|}^2|b{|}^2-(\overline{b}.\overline{c}{)}^2$

$=a^2(b.c)$

(If < b/w b and c is $\theta$ then $|a|=\lambda |b||c|\sin \theta$ and $(|a||b{|}^2=(b.c{)}^2)$

$=b^2{c}^2(1-{\cos }^2\theta )$

$=b^2{c}^2{\sin }^2\theta$