Question

If $\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ are non-coplanar vectors, then the vector $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})+(\overrightarrow{a}\times \overrightarrow{c})\times (\overrightarrow{d}\times \overrightarrow{b})+(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})$ is parallel to

• A. $\overrightarrow{a}$
• B. $\overrightarrow{b}$
• C. $\overrightarrow{C}$
• D. $\overrightarrow{a}+\overrightarrow{b}$

Answer 1

Answer: (A)

$\overrightarrow{a}$

we have,

$(\overline{a}\times \overline{b})\times (\overline{c}\times \overline{d})+(\overline{a}\times \overline{c})\times (\overline{d}\times \overline{b})+(\overline{a}\times \overline{d})\times (\overline{b}\times \overline{c})$
$=\left[\right(\overline{a}\times \overline{b}).\overline{d}]\overline{c}-\left[\right(\overline{a}\times \overline{b}).\overline{c}]\overline{d}+\left[\right(\overline{a}\times \overline{c}).\overline{b}]\overline{d}-\left[\right(\overline{a}\times \overline{c}).\overline{d}]\overline{b}+\left[\right(\overline{a}\times \overline{d}).\overline{c}]\overline{b}-\left[\right(\overline{a}\times \overline{d}).\overline{b}]\overline{c}$
$=2\left[\overline{a}\overline{d}\overline{c}\right]\overline{b}+2\left[\overline{d}\overline{a}\overline{b}\right]\overline{c}+2\left[\overline{a}\overline{c}\overline{b}\right]\overline{d}$
$=-2\left[\overline{b}\overline{c}\overline{d}\right]\overline{a}$

Hence the answer is parallel to $\overline{a}$.