Question

If $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, $\overrightarrow{d}$ are any four vectors then $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})$ is a vector

• A. Perpendicular to $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, $\overrightarrow{d}$
• B. Along the line of intersection of two planes, one containing $\overrightarrow{a}$, $\overrightarrow{b}$ and the other containing $\overrightarrow{c}$, $\overrightarrow{d}$
• C. Equally inclined to both $\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{c}\times \overrightarrow{d}$
• D. None of these

Answer 1

Answer: (B), (C)

The correct options are
B Along the line of intersection of two planes, one containing $\overrightarrow{a}$, $\overrightarrow{b}$ and the other containing $\overrightarrow{c}$, $\overrightarrow{d}$
C Equally inclined to both $\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{c}\times \overrightarrow{d}$

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

checking for option (A)

if dot product with $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}and\overrightarrow{d}$ gives 0 then correct or perpendicular

so taking dot product with $\overrightarrow{a}$

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

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

similarly invalid for vectors b,c,d

Cheking option (b)

eq of line of intersection of two planes

$\overrightarrow{r}=(x\hat{i}+y\hat{j}+z\hat{k})+\lambda (\overrightarrow{n_1}\times \overrightarrow{n_2})$

where $\overrightarrow{n_1},\overrightarrow{n_2}$ are normal vector of planes

SO if planes $P_{1}and{P}_2$ contains $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c},\overrightarrow{d}$

then $\overrightarrow{n_1}\times \overrightarrow{n_2}=(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})$

IT is valid

SO third option also valid above solution