Question

Let the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ be such that $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0$. Let $P_1$ and $P_2$ be planes determined by pair of vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c},\overrightarrow{d}$ respectively. Then the angle between $P_1$ and $P_2$ is

• A. $0^{\circ }$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $0^{\circ }$

A vector perpendicular to the plane $P_1$ of $\overrightarrow{a},\overrightarrow{b}$ is $\overrightarrow{a}\times \overrightarrow{b}$
A vector perpendicular to the plane $P_2$ of $\overrightarrow{c},\overrightarrow{d}$ is $\overrightarrow{c}\times \overrightarrow{d}$.
$\Rightarrow (\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0\Rightarrow (\overrightarrow{a}\times \overrightarrow{b})\parallel (\overrightarrow{c}\times \overrightarrow{d})$
$\therefore$ The angle between the planes is $0^{\circ }$.