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)

$\overrightarrow{a}$

Given expression

$(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})+(\overrightarrow{a}\times \overrightarrow{c})\times (\overrightarrow{d}\times \overrightarrow{b})+(\overrightarrow{a}\times \overrightarrow{d})\times (\overrightarrow{b}\times \overrightarrow{c})$

$\left(\right(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{d})\overrightarrow{c}-\left(\right(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{c})\overrightarrow{d}+\left(\right(\overrightarrow{d}\times \overrightarrow{b})\cdot \overrightarrow{a})\overrightarrow{c}-\left(\right(\overrightarrow{d}\times \overrightarrow{b})\cdot \overrightarrow{c})\overrightarrow{a}+\left(\right(\overrightarrow{b}\times \overrightarrow{c})\cdot \overrightarrow{a})\overrightarrow{d}-\left(\right(\overrightarrow{b}\times \overrightarrow{c})\cdot \overrightarrow{d})\overrightarrow{a}$

$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]\overrightarrow{c}-\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{d}+\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]\overrightarrow{c}-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}+\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{d}-\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}$

$2\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]\overrightarrow{c}-2\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}$

Let's check given expression is parallel to $\overrightarrow{a}$

$\overrightarrow{a}\times \left(2\right[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]\overrightarrow{c}-2[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\left]\overrightarrow{a}\right)=0$

$2\overrightarrow{a}\times \overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{d})\overrightarrow{c}-2\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]\overrightarrow{a}\times \overrightarrow{a})=0$

$0-0=0$

it is parallel to $\overrightarrow{a}$

Let's check given expression is parallel to $\overrightarrow{b}$

$\overrightarrow{b}\times \left(2\right[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]\overrightarrow{c}-2[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\left]\overrightarrow{a}\right)=0$

Here $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]and\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]$ are scalar

So, $\overrightarrow{b}\times (2p\overrightarrow{c}-2r\overrightarrow{a})=0$

$2p\overrightarrow{b}\times \overrightarrow{c}-2r\overrightarrow{b}\times \overrightarrow{a}\ne 0$

It is not parallel to $\overrightarrow{b}$